NEW  YORK; 


IN   MEMORIAM 
FLOR1AN  CAJORI 


2L-     s= 


/ 


2 

J      J 


t 


'£/  2- 


s 


•J  *) 


MENTAL  ARITHMETIC; 

OB, 

ORAL  EXERCISES  IN  ABSTRACT 


COMMERCIAL  ARITHMETIC, 

WITH 

FIRST  LESSONS  IS  WRITTEN  ARITHMETIC.) 

FOR   TUB 

USE  OF  SCHOOLS. 
BY    CHAS.  S.  TENABLE, 

»/ 

PROFESSOR  OP  MATHEMATICS  IN  THE  UNIVERSITY  OP   VIRGINIA*,  FORMERLY  PROFESSOR 
OF     MATHEMATICS     AND    ASTRONOMY,    UNIVERSITY    OF    SOUTH    CAROLINA;     FOR- 
MERLY   PROFESSOR   OF   NATURAL   PHILOSOPHY   AND    CHEMISTRY,    UNI- 
VERSITY OF   GEORGIA;    FORMERLY  PROFESSOR  OF   MATHEMA- 
TICS IN  HAMPDEN  SIDNEY   COLLEGE,   VIRGINIA. 


NEW  YORK: 

RICHARDSON    &    COMPANY, 

14  BOND  STREET. 

1867. 


MATHEMATICAL  SERIES. 


1.  First  Lessons  in  Numbers. — An  illustrated,  easy,  and  attractive 

Mental  Arithmetic,  for  the  use  of  Families  and  Primary  Schools. 
Adapted  to  the  OBJECT  METHOD.  30  cents. 

2.  Mental  Arithmetic. — After  the  inductive  method  of  Pestalozzi, 

with  first  lessons  in  Written  Arithmetic.  For  Common  Schools  and 
Academies.  45  cents. 

3.  Elements  of  Written  Arithmetic. — In  which   the  fundamental 

operations  on  Abstract  and  Denominate  Numbers,  and  their  various 
applications,  are  thoroughly  explained  and  illustrated  by  numerous 
examples  adapted  to  the  business  of  practical  life.  For  the  use  of 
Schools.  $1.00. 

4.  High  School  Arithmetic.     FOR  THE  PSE   OF  SCHOOLS  AND  COL- 

LEGES.— In  this  work  the  reasoning  on  the  fundamental  principles 
will  be  more  fully  developed,  and  illustrated  by  a  greater  variety 
and  number  of  applications  and  examples.  In  addition  to  the  sub- 
jects usually  treated  in  Text-books  of  Arithmetic,  it  will  contain 
several  chapters  on  Notation,  Theory  of  Numbers,  Progressions, 
Logarithms,  and  the  use  of  Logarithmic  Tables. 

5.  Algebra.    FOR  THE  USE  OF  SCHOOLS  ANI>  COLLEGES. — Containing  the 

latest  improvements.  To  a  thorough  discussion  of  the  subjects 
usually  treated  in  Textbooks  on  this  subject  will  be  added  chapters 
on  the  Theory  of  Determinants,  and  other  lessons  from  the  modern 
higher  Algebra,  for  the  use  of  University  Students. 

6.  Legendre's  Geometry* — To  the  complete  translation  of  the  latest 

edition  of  the  standard  work  of  Lespendre  will  be  appended  many 
exercises  in  Geometrical  Analysis,  with  hints  for  the  solution  of  the 
same.  Also,  notes  on  the  application  of  the  method  of  Limits  to 
the  demonstration  of  theorems  in  Elementary  Geometry. 

7.  Analytic   Geometry,— Part  First— Plane    Co-ordinate    Geometry, 

The  Point,  Straight  Line,  Conic  Sections,  some  of  the  hisrher  Plane 
Curves,  Algebraic  and  Transcendental.  Also,  the  Method  of 
Abridged  Notation. 

Part  Second— The  Point  and  Straight  Line  in  Space,  the  Plane, 
Surfaces  of  the  Second  Order,  Cones,  Cylinders,  and  Surfaces  of 
Revolution,  Conoids,  Helicoids,  Curves  of  Double  Curvature,  Ana- 
lytical Theory  of  Projections. 

RICHARDSON  &  CO.,  PUBLISHERS,  NEW  YORK. 

Entered  according  to  Act  of  Congress,   in  the  year  1866, 

BY  RICHARDSOX  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the 
Southern    District  of  New  York. 


A 


CONTENTS. 


CHAPTER    I. 

PAGE 

Formation  of  Numbers  by  Counting — Addition — Use  of  the  Arabic 

Figures 7 


CHAPTER    II. 
Subtraction 20 

CHTAPTER    III. 
Multiplication 30 

C3APTER    IT. 
Division 40 

CHAPTER    V. 
Miscellaneous  Questions 59 

CHAPTER    VI. 
Multiples,  Divisors,  and  Prime  Factors 67 

CHAPTER    VII. 
Fractions — Proportion 72 


IV  CONTENTS. 

CHAPTER     VIII. 

PAGB 

Denominate  Numbers 119 

CHAPTER    IX. 
The  Metric  System  of  Weights  and  Measures 132 

CHAPTER    X. 
Percentage — Interest 1ST 

CHAPTER    XI. 
Miscellaneous  Problems 151 

9 

CHAPTER    XII. 
Elementary  Lessons  in  Written  Arithmetic 1GO 


PREFACE. 


THE  advantages  of  the  study  of  Mental,  or  Oral,  Arithmetic 
are  very  generally  acknowledged,  and  yet  it  is  much  neglected 
in  the  schools.  The  present  work  is  prepared  as  a  companion 
to  those  treatises  on  written  Arithmetic  which  are  not  mere 
collections  of  examples,  but  whose  aim  is  to  place  the  study  in 
its  proper  position  among  other  branches  of  a  rational  education. 

Great  care  has  been  taken  to  arrange  the  subjects  and  to 
select  and  frame  the  questions  in  such  a  manner  that  the  pupil 
may  not  only  acquire  readiness  and  accuracy  in  the  solution  of 
numerical  problems,  but  may  gain  a  thorough  knowledge  of  all 
the  simpler  relations  among  numbers,  and  learn  the  reason  for 
each  step  taken  in  the  various  operations  performed  on  them. 

The  method  pursued  is  that  identified  with  the  name  of  Pes- 
talozzi.  The  clearest  exposition  of  this  method  in  abstract 
numbers  is  to  be  found,  perhaps,  in  Reiner's  Lessons  in  a  Pesta- 
lozzian  school  in  Surrey,  England.  Use  has  been  made  of  this, 
and  of  Menzel's  valuable  G-erman  collection  of  examples  for 
"  Head-reckoning." 

The  attention  of  teachers  is  called  to  the  following  points  in 
this  little  work :  Its  order  and  method ;  the  chapters  on  mul- 
tiples, divisors,  and  prime  factors ;  the  treatment  of  vulgar  frac- 
tions ;  the  mode  of  teaching  proportional  numbers ;  the  brief  and 
direct  analysis  of  the  various  solutions  of  questions  given  ;  the 
tables  of  French  decimal  weights  and  measures  (now  authorized 
in  this  country  by  the  United  States  Government),  with  appro- 
priate questions,  illustrating  the  superiority  of  these  tables  over 
tho  ordinary  ones  for  ready-reckoning. 


Yl  PREFACE. 

A  few  pages  of  written  Arithmetic  are  appended,  for  the 
convenience  of  those  who  may  wish  to  acquire  some  familiarity 
with  the  operations'  on  higher  abstract  numbers  before  entering 
upon  the  next  book  of  the  series.  These  pages  may  also  be 
used  to  advantage  concurrently  with  the  rest  of  the  book. 


SUGGESTIONS  TO  TEACHERS. 

1.  Require  the   pupil  to  commit  to  memory  the  necessary 
tables  of  Addition,  Subtraction,  Multiplication,  and  Division. 

2.  Practice  the  pupil  frequently  in  counting  forwards  and 
backwards  by  twos,  threes,  tens,  &c.,  &c.     This  will  cause  him 
to  understand  thoroughly  the  basis  of  all  the  operations  on 
whole    numbers,    and    that    Multiplication    and   Division   are 
abridged  forms  of  Addition  and  Subtraction. 

3.  Do  not  enter  upon  the  subject  of  fractions  until  the  pupil 
is  very  familiar  with  the  questions  on  factors,  multiples,  and 
divisors. 

4.  Add  examples  and  illustrations  to  those  in  the  book,  and 
require  your  pupils  in  each  lesson  to  frame  questions  and  give 
their   solutions.     This  will   be   found  a  most  useful  exercise, 
especially  in  the  miscellaneous  questions  on  Proportion  and  its 
applications. 

5.  Pupils  in  written  Arithmetic  should  be  called  on  to  solve, 
orally,  questions  similar  to  those  in  each  .set  of  examples  which 

.  they  work  on  the  slate. 


MENTAL    ARITHMETIC, 


CHAPTER    I. 

ADDITION. 
LESSON    I. 

1.  John  has  one  knife ;    if  his  father  gives  him 
another,  how  many  will  he  have  ? 

2.  Frank  has  two  oranges,  and  Charles  one ;  how 
many  have  they  both  together  ? 

3.  James  is  in  school  three  hours  every  morning, 
and  one  hour  each  evening ;  how  many  hours  is  he  in 
school  each  day  ? 

4.  A  boy  took  four  eggs  from  one  bird's  nest,  and 
one  from  another ;  how  many  birds'  eggs  did  he  take  ? 

5.  Mary  has  five  tulips  in  bloom  on  one  bed  in  the 

farden,  and   one  upon    another ;    how   many   tulips 
loom  on  both  beds  ? 

6.  One  and  one  are  How  many  ?    Two  and  one  ? 
Three  and  one  ?     Four  and  one  ?     Five  and  one  ?     Six 
and  one?     Seven  and  one  ^  Eight  and  one?     Nine 
and  one  ?     Ten  and  one  ? 

7.  Count  from  one  to  ten.     Count  the  birds  you 
know.     Count  the  flowers  you  know.     Count  from  ten 
to  twenty.     Count  the  number  of  words  in  the  first  two 
questions  of  this  lesson.     Count  from  twenty  to  thirty. 
Count  from  thirty  to  forty. 

8.  Begin  at  twenty  and  add  successively  the  letters 
of  the  sentence,  "  Labor  overcomes  the  greatest  diffi- 


8  ADDITION. 

culties."  Count  from  fifty  to  sixty.  From  sixty  to 
seventy,  eighty,  ninety,  one  hundred.  Commence  at 
the  first  word  of  this  lesson,  and  count  the  words  as 
far  as  one  hundred. 

9.  How  many  tens  have  you  counted  in  counting 
twenty  ?   in   counting    thirty  ?    forty  ?    fifty  ?    sixty  ? 
seventy  ?  eighty  ?  ninety  ?  one  hundred  ? 

10.  Count  from  one  hundred  to  two  hundred. 

11.  How  would  you  count  from  two  hundred  to 
three  hundred  ? 

ANS. — By  counting  from  one  to  one  hundred^  and 
repeating  two  hundred  before  each  number  counted. 

12.  How  would  you  count  from  three  hundred  to 
four  hundred  ?    to  five  hundred  ?    six  hundred  ?  seven 
hundred?    eight  hundred?   nine  hundred?    ten  hun- 
dred? 

13.  What  do  we  call  ten  hundred  ? 
ANS. — One  thousand. 

LESSON    II. 

1.  A  man  has  fifty-two  white  sheep  and  one  black 
one;  how  many  has  he  altogether? 

2.  One  company  of  soldiers  has  sixty-five  men  in 
it,  a  second  company  has  one  man  more ;  how  many 
men  in  the  second  company  ? 

3.  Of  a  flock  of  blackbirds,  forty-five  are  on  a  tree, 
and  one  on  the  fence :  how  many  blackbirds  in  the 
flock? 

4.  Ten  and  one  are  how  many?     Twenty  and  one  ? 
Thirty  and  one  ?     Forty  and  one  ?     Fifty  and  one  ? 
Seventy  and  one  ?     Ninety  and  one  ? 

5.  Eleven  and  one  are   how  many  ?     Twenty-one 
and  one  ?     Thirty-one  and  one  ?     Fifty-one  and  one  ? 
Sixty-one  and  one  ?     Eighty -one  and  one  ?     One  hun- 
dred and  one  and  one  ? 

6.  Twelve  and  one  are  how  many  ?     Twenty-two 


ADDITION.  9 

and  one  ?  Thirty-two  and  one  ?  Forty-two  and  one  ? 
Fifty-two  and  one?  Sixty-two  and  one?  Seventy- 
two  and  one  ?  Eighty -two  and  one  ?  Ninety-two 
and  one  ? 

7.  Fourteen    and    one?      Twenty-four    and    one? 
Thirty- four  and  one  ?    Forty-four  and  one  ?    Fifty-four 
and  one?    Sixty-four  and  one  ?    Seventy-four  and  one  ? 
Eighty-four  and  one  ?     Ninety-four  and  one  ? 

8.  Fifteen  and  one  are  how  many  ?     Twenty-five 
alid  one  ?  &c. 

9.  Nineteen  and  one  are  how  many  ?     Twenty-nine 
and  one?  &c. 

10.  One  ten  and  one  ten  are  how  many  ?     Two  tens 
and  one  ten  ?     Three  tens  and  one  ten  ?     Seven  tens 
and  one  ten?     Eight  tens  and  one  ten?     Ten  tens  and 
one  ten  ? 

11.  One  hundred  and  one  hundred  are  how  many? 
Two  hundred  and  one  hundred  ?     Three  hundred  and 
one  hundred?      Four  hundred   and    one    hundred? 
Nine  hundred  and  one  hundred  ? 


LESSON    III. 

1.  Two  oranges  and  two  oranges  are  how  many 
oranges  ? 

2.  A  planter  made  three  bales  of  cotton  on  one  acre 
of  land,  and  two  on  another  acre ;   how  many  bales 
did  he  make  on  both  ?  . 

3.  Charles  has  three  marbles,  and  James  has  three ; 
how  many  have  they  together  ? 

4.  A  traveller  walked  three  miles  the  first  hour,  and 
four  miles  the  second  ;  how  many  miles  did  he  walk 
in  the  two  hours  ? 

5.  A  planter  makes  five  hogsheads  of  tobacco  one 
year,  and  four  the  next ;  how  many  hogsheads  in  the 
two  years  ? 

i* 


10  ADDITION. 

6.  One  and  two  are  how  many  ?     One  and  three  ? 
One  and  four  ? 

7.  Two  and  two  ?     Two  and  three  ?     Two  and  four  ? 

8.  Three  and  two?     Three  and  three?     Three  and 
four? 

9.  Four  and  two  ?     Four  and  three  ?     Four  and 
four? 

10.  Five  and  two  ?     Five  and   three  ?     Five   and 
four? 

11.  Six  and  two?     Six  and  three?     Six  and  four? 

12.  Seven  and  two  ?     Seven  and  three  ?     Seven  and 
four? 

13.  Eight  and  two  ?     Eight  and  three  ?     Eight  and 
four  ? 

14.  Nine  and  two  ?     Nine  and  three  ?     Nine  and 
four  ? 

15.  Ten  and  two  ?     Ten  and  three  ?     Ten  and  four  ? 

16.  Twenty  and  two  are  how  many  ?     Thirty  and 
two?     Forty  and  two?     Fifty  and  two?     Sixty  and 
two  ?     Seventy  and  two  ?     Eighty  and  two  ?     Ninety 
and  two  ? 

17.  Twenty  and  three  are  how  many  ?  &c. 

18.  Twenty  and  four  are  how  many?  &c. 

The  teacher  will  cause  the  pupil  to  go  through  the  same  addition  as 
above  with  three  and  four. 

i 

19.  Eleven  and  two  are  how  many?     Twenty-one 
and  two  ?     Thirty-one  and  two  ?     Forty-one  and  two  ? 
Fifty-one  and  two  ?     Sixty-one  and  two  ?     Seventy- 
one  and  two  ?     Eighty-one  and  two  ?     Ninety-one  and 
two? 

20.  Eleven  and  three  are  how  many  ?     Twenty -one 
and  three  ?  &c. 

21.  Eleven  and  four  are  how  many?     Twenty-one 
and  four?  &c. 

22.  Twelve  and  two  are  how  many  ?    .Twenty-two 
and  two  ?  &c. 


ADDITION.  11 

23.  Twelve  and  three  are  how  many  ?     Twenty-two 
and  three  ?  &c. 

24.  Twelve  and  four  are  how  many  ?     Twenty -two 
and  four  ?  &c. 

25.  Thirteen  and  two  ?    Twenty-three  and  two  ?  &c. 

26.  Thirteen  and  three  ?     Twenty -three  and  three? 
&c. 

27.  Thirteen  and  four?  .    Twenty-three  and  four? 
&c. 

28.  Fourteen  and  two  ?     Twenty-four  and  two  ?  &c. 

29.  Fifteen  and  two  ?     Twenty-five  and  two  ?  &c. 

30.  Sixteen  and  two  ?     Twenty-six  and  two  ?  &c. 

31.  Seventeen  and  two  ?     Twenty-six  and  two  ?  &c. 

32.  Eighteen  and  two  ?    Twenty-eight  and  two  ?  &c. 

33.  Nineteen  and  two  ?     Twenty-nine  and  two  ?  &c. 

The  teacher  should  extend  these  questions  as  far  as  ninety-three  in, 
the  same  manner  with  those  which  are  completed,  and  make  the 
same  series  of  additions  with  three  and  four. 

34.  Count  by  twos  to  one  hundred  and  one,  begin- 
ning with  one. 

35.  Count  by  twos  to  one  hundred,  beginning  with 
two. 

36.  Count  by  threes  to  one  hundred, -beginning  with 
one. 

37.  Count  by  threes  to  one  hundred  and  one,  begin- 
ning with  two. 

38.  Count  by  threes  to  ninety-nine,  beginning  with 
three. 

39.  Eighty-seven  and  one,  and  two,  and  three,  are 
how  many  ? 

40.  Ninety  two  and  two,  and  three,  and  four,  are 
ow  many? 

41.  Thirty-five  and  two,  and  three?      Forty-three 
,nd    four,    and    two?       Fifty-four    and    four,    and 

three  ? 


12 


ADDITION. 


LESSON    IV. 

We  have  thus  far  written  out  the  names  of  the 
different  numbers.  For  the  sake  of  abbreviation  in 
the  succeeding  lessons,  we  will  use  the  figures  of  the 
Arabic  notation. 

These  are — 

one    two    three    four    five    six     seven     eight    nine   zero  or  naught 
1234567  89  0 

NOTE. — For  a  more  detailed  explanation  of  the  use  of  figures,  see 
first  lessons  in  Numbers,  page  31. 

We  have  seen,  in  counting,  there  are  two  tens  in 
twenty,  three  in  thirty,  four  in  forty,  &c.,  and  ten 
tens  in  one  hundred.  If  now  we  wish  to  express 
numbers  greater  than  nine  by  those  figures,  we  may 
do  so  readily  by  the  use  of  columns  and  the  0 
(zero). 


ADDITION.  13 

Hundreds.     Tens.     Units  or  ones. 

5  6 

4  0 

123 
100 

Thus  5  in  the  column  of  tens  and  6  in  the  column 
of  ones  or  units,  means  five  tens  and  six  units,  or  fifty- 
six  ;  4  in  the  column  of  tens  and  0  in  the  column  of  units 
indicates  four  tens  and  no  units,  of  forty ;  1  in  the 
column  of  hundreds,  and  2  in  the  column  of  tens,  and  3 
in  that  of  units,  indicates  one  hundred,  two  tens,  and 
three  units,  or  one  hundred  and  twenty-three ;  1  in 
the  hundreds  column,  0  in  tens,  and  0  in  units,  indi- 
cates one  hundred.  We  leave  out  the  columns  in  the 
actual  writing  of  numbers  and  write  them  thus :  56, 
40,  123,  100,  and  simply  recollect  the  local  values  of 
the  figures. 

1.  What  is  the  local  value  of  r  the  figure  on  the  right 
of  a  number  ? 

ANS. —  Units. 

2.  What  is  the  local  value  of  the  figure  next  to  the 
right-hand  figure  ? 

3.  The  third  figure  from  the  right  ?  the  fourth  ? 

4.  What  is  the  meaning  and  use  of  the  0? 

NOTE. — The  pupil  should  now  be  required  to  read  the  following 
table,  and  be  drilled  in  the  use  of  figures,  until  he  can  readily  write  any 
number  from  one  to  one  thousand  in  figures: 

10  20  40  30  88 

11  21  41  31  90 

12  22  42  32  •   93 

13  23  43  60  99 

14  24  44  61  100 

15  25  45  62  110 

16  26  50  70  101 

17  27  51  78  255 

18  28  52  80  999 

19  29  53  81  1000 


14 


ADDITION. 


LESSON    V. 


1.  John  bought  a  lemon  for  five  cents,  and  an  apple 
for  five  cents ;  how  much  did  they  both  cost  him  ? 

2.  One  man  rides  six  miles  in  one  hour,  and  another 
rides  three  miles  more  than  the  first ;  how  many  miles 
does  the  latter  ride  in  one  hour? 

3.  Edward  begins  to  work   at   six  o'clock  in  the 
morning  and  works  five  hours ;  at  what  o'clock  does 
he  stop,  work  ? 

4.  Harry  has  seven  peaches,  and  Frank  has  five ; 
how  many  peaches  have  both  ? 

5.  A  farmer  has  seven  cows  in  one  pasture,  and  nine 
in  another ;  how  many  has  he  in  both  ? 

1  and  7  \ 


6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 

1 

2 
3 
4 
5 

6 
7 
8 
9 

and 
and 
and 
and 
and 
and 
and 
and 
and 

5  are  how 
5  ?     2  and 
5  ?     3  and 
5  ?     4  and 
5  ?     5  and 
5  ?     6  and 
53     7  and 
5  ?     8  and 
5  ?     9  and 

many  \ 
6?     2 
6?     3 
6?     4 
6?     5 
6?     6 
6?    7 
6?     8 
6?     9 

1 
and 
and 
and 
and 
and 
and 
and 
and 

and  6? 

7? 
7? 
7? 
7? 
7? 
7? 
7.1 
7? 

15.  10  and  5  are  how  many  ?  20  and  5  ?     30  and 
5  ?     40  and  5  ?     50  and  5  ?     60  and  5  ?     70  and  5  2 
80  and  53     90  and  5  ? 

16.  10  and  6  are  how  many  ?  20  and  6  ?  &c. 

17.  10  and  7  are  how  many  ?  20  and  7  ?  &c. 

18.  11  and  5  are  how  many  ?  21  and  5  ?  &c. 

19.  11  and  6  are  how  many  ?  21  and  6  ?  &c. 

20.  11  and  7  are  how  many  ?  21  and  7  ?  &c. 
,    21.  12  and  5  are  how  many  ?  22  and  5  ?  &c. 

22.  12  and  6  are  how  many  ?  22  and  6  ?  &c.      . 

23.  12  and  7  are  how  many  ?  22  and  7  ?  &c. 

24.  13  and  5  are  how  many  ?  23  and  5  ?  &c. 

25.  13  and  6  are  how  many  ?  23  and  6  ?  &c. 


ADDITION. 


15 


26.  13  and  7  are  how  many  ?  23  and  7  ?  &c. 

27.  14  and  5  are  how  many  ?  24  and  5  ?  &c. 

28.  14  and  6  are  how  many  ?  24  and  6  ?  &c. 

29.  14  and  7  are  how  many  ?  24  and  7  ?  &c. 

30.  15  and  5  are  how  many  ?  25  and  5  ?  &c. 

31.  16  and  5  are  how  many  ?  26  and  5  ?  &c. 

32.  17  and  5  are  how  many  ?  27  and  5  ?  &c. 

33.  18  and  5  are  how  many  ?  28  and  5  ?  &c. 

34.  19  and  5  are  how  many  ?  29  and  5  ?  &c. 

To  be  carried  out  as  far  as  91,  92,  93,  94,  &c.,  by  the  pupil,  for  5,  6, 
and  7. 

LESSON    VIC 

I.1  Count  by  5's  to  101,  beginning  with  1.  To  102, 
beginning  with  2.  To  103,  beginning  with  3.  To 
104,  beginning  with  4.  To  100,  beginning  with  5. 

2.  Count  by  6's,  beginning  with  1,  2,  3,  4,  5,  6,  re- 
spectively.    Thus,  1,  7,  13,  &c.     2,  8, 14,  &c.,  &c. 

3.  Count  by  7's,  beginning  with  1,  2,  3,  4,  5,  6,  re- 
spectively.    Thus,  1,  8,  15,  &c.     2,  9,  16,  &c.,  &c. 

4.  A  man  had  sixty-five  sheep  in  one  fold,  five  in 
a  second,  six  in  a  third,  and  seven  in  a  fourth ;  how 
many  had  he  altogether  ? 

5.  A    farmer  gathered   fifty-six  barrels  of   apples 
from  his  orchard,   and  seven  barrels  from  two  trees 
which  stood  in  the  corner  of  his  yard ;  how  many  bar- 
rels did  he  gather  ? 

6.  There  are  three   hundred  soldiers  in  one  regi- 
ment, and  six  hundred  in  another  ;  how  many  in  the 
two  regiments? 

7.  2  tens  and  5  tens  are  how  many  ?     2  tens  and  6 
tens  ?     2  tens  and  7  tens  ? 

8.  9  tens  and  5  tens  are  how  many  ?     9  tens  and 
6  tens  ?     9  tens  and  7  tens  ? 

9.  3  hundred  and  5  hundred  are  how  many  ?      3 
hundred  and  6  hundred  ?     3  hundred  and  7  hundred  ? 


16  ADDITION. 

10.  4  hundred  and  5  hundred  are  how  many  ?     4 
hundred  and  6  hundred  ?     4  hundred  and  7  hundred  ? 

11.  Add  fifty-seven,  seven,  six,  and  five. 

12.  Add  forty-nine,  seven,  five,  three,  and  four. 

13.  How  many  are  ninety-seven,  six,  seYen,  four, 
seven,  and  three? 

14.  How  many  are  forty-five,  six,  three,  five,  six, 
four,  three,  and  four  ? 

LESSON    VII. 

1.  Frank  is  seven  years  old,  his  sister  is  eight  years 
older ;  how  old  is  his  sister  ? 

2.  Thomas  killed  six  squirrels  on  Monday,  and  nine 
on  Tuesday  ;  how  many  squirrels  did  he  kill  on  both 
days? 

3.  A  gardener  gathered  five  bushels  of  apples  from 
one  tree,  and  ten  bushels  from  another ;  how  many 
bushels  from  the  two  trees  ? 

4.  1  and  8  are  how  many  ?     1  and  9  ?     1  and  10  ? 

5.  2  and  8  ?     2  and  9  ?   "2  and  10  ? 

6.  3  and  8?     3  and  9  ?     3  and  10? 

7.  4  and  8  ?     4  and  9  ?     4  and  10? 

8.  5  and  8  ?     5  and  9  ?     5  and  10  ? 

9.  6  and  8  ?     6  and  9  ?     6  and  10  ? 

10.  7  and  8?     7  and  9?     7  and  10? 

11.  8  and  8  ?     8  and  9  ?     8  and  10  ? 

12.  9  and  8?     9  and  9  ?     9  and  10  ? 

13.  10  and  8  are  how  many  ?    20  and  8  ?    30  and  8  ? 
&c.,  to  98. 

14.  11  and  9  are  how  many  ?     21  and  9  ?     81  and  9  ? 
&c.,  to  10. 

15.  12  and  10  are  how  many  ?     22  and  10  ?     32  and 
10  ?  &c.,  to  102. 

16.  13  and  8  are  how  many  ?    23  and  8  ?    33  and  8  ? 
&c.,  to  101. 

17.  14  and  9  are  how  many  ?    24  and  9  ?     34  and  9  ? 
&c.,  to  103. 


ADDITION.  IT 

18.  15  and  10  are  how  many?     25  and  10  ?     35  and 
10  ?  &c.,  to  105. 

19.  1 6  and  8  are  how  many  ?     26  and  8  ?     36  and  8? 
&c.,  to  104. 

20.  17  and  9  are  how  many  ?    27  and  9  ?     37  and  9  ? 
&c.,  to  106. 

21.  18  and  10  are  how  many  ?     28  and  10  ?     38  and 
10  I  &c.,  to  108. 

22.  19  and  8  are  how  many  ?     29  and  8  ?    39  and  8  ? 
&c.,  to  107. 

23.  10  and  10  are  how  many  ?     20  and  10  ?     30  and 
10?  &c.,  to  100. 

24.  How  much  is  forty-five  and  eight  ? ,      Two  and 
seven  and  five  and  eight  and  nine  are  how  many  ? 

25.  Fifty-seven  and  eight  are  how  many  ? 

26.  Ninety-three  and  nine  ?  one  hundred  and  sixteen 
and  ten  ?     5,  9,  6,  9,  3,  8,  7,  2,  and  1  are  how  many? 
Ninety-seven,  eight,  nine,  and  ten  are  how  many  ? 

LESSON    VIII. 

1.  Count  by  8's,  beginning  with  1,  2,  3,  4,  5^6,  7,  and 
8  respectively.     Thus,  1,  9,  17,  &c.     2,  10.  18,  &c., 
&c. 

2.  Count  by  9's,  beginning  with  1,  2,  3,  4,  5,  6,  7, 
8,  and   9  respectively.     Thus,   1,  10,  19,  &c.     2,  11, 
20,  &c.,  &c. 

3.  Count  by  10's,  beginning  with  1,  2,  3,  4,  5,  6,  7, 
8,  9,  and  10  respectively.     Thus,  1,  11,  21,  &c.     2, 
12,  22,  &c.,  &c. 

4.  A  farmer  sends  sheep  to  market  in  two  droves, 
one  drove  has  twenty  sheep  in  it  and  the  other  eighty  ; 
how  many  sheep  in  both  droves  ? 

5.  A  man  makes  five  hundred  bushels  of  wheat  on 
one  farm,  and  nine  hundred  on  another ;  how  many 
hundred  bushels  does  he  make  on  both  farms? 

6.  A  grazier  has  one  hundred  and  seventeen  head  of 


18  ADDITION. 

cattle  in  one  field,  and  ten  head  in  another  ;  how  many 
head  in  the  two  fields  ? 

7.  3  tens  and  8  tens  are  how  many  tens  ?     3  tens 
and  9  tens  ?     3  tens  and  10  tens  ? 

8.  8  tens  and  8  tens  are  how  many  tens  ?     8  tens 
and  9  tens?     8  tens  and  10  tens ? 

9.  2  hundred  and  8  hundred  are  how  many  hun- 
dred ?    2  hundred  and  9  hundred  ?     2  hundred  and  10 
hundred  ? 

10.  7  hundred  and  8  hundred  ?     7  hundred  and  9 
hundred  ?     7  hundred  and  10  hundred  \ 


LESSON    IX. 

1.  A  father  has  three  books,   and  wishes  to  give 
them  to  his  three  sons ;  how  many  would  he  give  to 
each  son? 

2.  If  he  wished  to  give  them  to  two  sons,  how  many 
would  he  give  to  one  and  how  many  to  the  other? 

3.  Three  can  be  broken  up  how  ? 

ANS. — Into  three  ones,  or  into  two  and  one. 

4.  4:  can  be  separated  into  what  numbers  ? 
ANS. — 3  and  1,  2  and  2,  1  and  1  and  1  and  1. 

5.  What  different  numbers  added  together  make 
5? 

6.  A  boy  has  six  marbles ;  into  what  different  heaps 
can  he  distribute  them  ? 

7.  6  can  be  separated  into  what  numbers  ? 

8.  7  can  be  separated  into  what  numbers  ? 

9.  What   different    numbers   added   together  will 
makeS?     9?     10? 

When  a  number  is  greater  than  ten,  and  less  than  one  hundred,  the 
inost  usual  and  useful  separation  in  our  system  of  numbers  is  into 
tens  and  units  ;  or,  if  greater  than  one  hundred,  and  less  than  a  thou- 
sand, into  hundreds,  tens,  and  units.  The  notation  or  mode  of  writing 
numbers,  which  has  been  explained  in  a  previous  lesson,  makes  this 
separation  very  easv. 

18 


ADDITION.  1 9 

10.  Separate  11,  12,  13,  14,  15,  27,  49,  58,  and  97 
into  tens  and  units. 

11.  Separate  137,  142,  156,  207,  320,  197  into  hun- 
dreds, tens,  and  units,  and  into  tens  and  units. 

12.  25  and  19  are  how  many? 

PROCESS. — Separate  19,  and  we  have  1  ten  and  9 
units  ;  25  and  10  are  35 ;  35  and  9  are  44.  Hence,  25 
and  19  are  44. 

13.  34  and  14  are  how  many  ?     31  and  18  ?     32  and 
17  ?     32  and  16  ? 

14.  14  and  19  ?     95  more  13  ?     46  and  12  ? 

15.  11  and  11  are  how  many  ?     22  and  11  ?     33  and 
11? 

16.  Count  by  ll's  to  99,  beginning  with  11. 

17.  12  and  12  are  how  many  ?     24  and  12  ?     36  and 
12? 

18.  Count  by  12's  to  96,  beginning  with  12. 

19.  40  and  20  are  how  many  ?     50  and  30  ?     60  and 
40  ?     30  and  60  ?     70  and  20  ?     90  and  50  ? 

20.  40  and  27  are  how  many  ? 

PROCESS. — 27  ^  20  andl;  40  and  20  are  60,  60  and 
7  make  67. 

21.  How  much   is  60   and  32?     70  and  29?     90 
and  67  ?     50  and  48  ?     80  and  22  ? 

22.  47   and  20  are  how  many?     58  and  30?     65 
and  40  ?     33  and  30  ?     72  and  20  ? 

23.  48  arid  27  are  how  many  ? 

PROCESS. — 40  and  20  are  60;  8  and  7  are  15;  60 
and  15  are  75.  Or  thus,  27  is  20  and  7,  48  and  20 
make  68,  68  and  7  make  75. 

24.  72  and  25  are  how  many !     64  and  56  ?     31 
and  22  ?     48  and  63  ?     84  and  26? 


20  SUBTRACTION. 

CHAPTEE    II. 

SUB  TRACTION. 
LESSON    I. 

1.  A  boy  has  one  apple,  and  gives  it  away  ;  how 
many  has  he  left  ?     Another  has  twenty  apples  and 
gives  them  away ;  how  many  has  he  left  ? 

2.  1  from  1  leaves  how  many  ?     20  from  20  leaves 
how  many  ? 

3.  Any  number  taken  from  itself  leaves  what  ? 

4.  A  boy  has  five  peaches,  and  eats  one  of  them  ; 
how  many  has  he  left  ? 

5.  Mary  had  five  books,  but  she  has  lost  two  of 
them  ;  how  many  has  she  now? 

6.  "William  had  ten  canary  birds  in  a  cage,  but  al- 
lowed two  of  them  to  escape  ;  how  many  has  he  left  ? 

7.  What  do  you  call  the  result  obtained  by  adding 
one  number  to  another  ? 

ANS. — Their  sum. 

8.  What  do  you  call  the  result  obtained  by  taking 
one  number  from  another  ? 

ANS  . — Their  differ  en  ce. 


9.  1  from    1  leaves  how  much  ? 

2  less  2  is  ? 

10.  1  from    2  leaves  ? 

3  less  2  ? 

11.  1  from    3  leaves? 

4  less  2  ? 

12.  1  from    4  leaves? 

5  less  2  ? 

13.  1  from    5  leaves  ? 

6  less  2  ? 

14.  1  from    6  leaves  ? 

7  less  2  ? 

15.  1  from    7  leaves  ? 

8  less  2  ? 

16.  1  from    8  leaves  ? 

.9  less  2  ? 

17.  1  from    9  leaves? 

10  less  2  ? 

18.  1  from  10  leaves  ? 

19.  11  less  1  is  ?     21  less  1  ?     31 

less  1  ?     41  less  1  ? 

51  less  1?    61  less  1?    71  less  1?    81 

lessl?     91  less  1? 

SUBTRACTION.  21 

20.  11  less  2  ?     21  less  2  ?  &c.,  to  101. 

21.  22  less  1  ?     23  less  1  ?     24  less  1  ?     35  less  1  ? 
46  less  1  ? 

22.  57  less  one?     68  less  1?     79  less  1  ?    110  less  1  ? 

23.  12  less  2  is  how  much?     22  less  2?     &c.,  to 
102. 

24.  13  less  2  ?     23  less  2  ?  &c.,  to  103. 

25.  14  less  2  ?     24  less  2  ?  &c.,  to  94. 

26.  15  less  2  ?     25  less  2  ?  &c.,  to  95. 

27.  16  less  2  ?     26  less  2  ?  &c.,  to  96. 

28.  18  less  2?     28  less  2?  &c.,  to  98. 

29.  19  less  2?     29  less  2?  &c.,  to  99. 

30.  Begin  at  100,  and  count  backward  down  to  1. 

31.  Begin  with  100,  and   count  backwards  by  2's 
down  to  2. 

32.  Begin  with    99,  and    count  backwards  by  2's 
down  to  1. 

33.  1  ten  from  2  tens  leaves  how  many?     1  ten 
from  3  tens? 

34.  1  ten  from  6  tens  ?     2  tens  from  8  tens  ? 

35.  1  hundred  from  4  hundred  leaves  how  many  ? 
1  hundred  from  8  hundred  ?     2  hundred  from.  7  hun- 
dred I    2  hundred  from  9  hundred  3 

LESSON    II. 

1.  George   had  ten  chickens,  butya  weasel  killed 
three  of  them ;  how  many  remained  ? 

2.  One  farmer  has  seven  horses  and  another  has 
four ;  how  many  more  has  the  first  than  the  second  ? 

3.  A  man  planted  twelve  trees,  but  five  of  them 
died  ;  how  many  lived  ? 

4.  A   man  owing  thirteen  dollars,  paid   three  of 
them  ;   how  much  did  he  still  owe  ? 

5.  A  farmer  had  fifteen  cows  in  one  pasture,  and  five 
in  a  second  ;    how  many  more  were  in  the  first  pas- 
ture than  in  the  second  'I 


SUBTRACTION. 


6. 
7. 
8. 
9. 
10. 
11. 
12. 
13. 
14. 
15. 

3  less 
4  less 
5  less 
6  less 
7  less 
8  less 
9  less 
10  less 
11  less 
12  less 

3 
3? 
3? 
3? 
3? 
3? 
3? 
3? 
3? 
3? 

4 
5 
6 
7 
8 
9 
10 
11 
12 
13 

less 
less 
less 
less 
less 
less 
less 
less 
less 
less 

4? 

4? 
4? 
4? 
4? 
4? 
4? 
4? 
4? 
4? 

5  less  5  ? 
6  less  5  ? 
7  less  5  ? 
8  less  5  ? 
9  less  5  ? 
10  less  5  ? 
11  less  5  ? 
12  less  5  ? 
13  less  5  ? 
14  less  5  ? 

16. 

13 

less 

3? 

14 

less 

4? 

15  less  5  ? 

17. 

10 

less 

3? 

20 

less 

3? 

30 

less 

3? 

&c., 

to  90. 

18. 

11 

less 

3? 

21 

less 

3? 

31 

less 

3? 

&c., 

to  91. 

19. 

12 

less 

3? 

22 

less 

3? 

32 

less 

3? 

&c., 

to  92. 

20. 

13 

less 

3? 

23 

less 

3? 

33 

less 

3? 

&c., 

to  93. 

21. 

14 

less 

3? 

24 

less 

3? 

34 

less 

3? 

&c., 

to  94. 

22. 

15 

less 

3? 

25 

less 

3? 

35 

less 

3? 

&c., 

to  95. 

23. 

16 

less 

3? 

26 

less 

3? 

36 

less 

3? 

&c., 

to  96. 

24. 

17 

less 

3? 

27 

less 

3? 

37 

less 

3? 

&c., 

to  97. 

25. 

18 

less 

3? 

28 

less 

3? 

38 

less 

3? 

&c., 

to  98. 

26. 

19 

less 

3? 

29 

less 

3? 

39 

less 

3? 

&c., 

to  99. 

NOTE. — The  teacher  should  cause  the  pupil  to  go  through  several  of 
the  above  subtractions  with  4  and  5  respectively. 

27.  Count  by  3's  backwards  from  100  to  1. 

28.  Count  by  4's  backwards  from  100  to  0. 

29.  Count  by  5 's  backwards  from  100  to  0. 

30.  3  tens  from  8  tens  leaves  how  many  ?     4  tens 
from  7  tens  ?     5  tens  from  9  tens  ? 

31.  One  veteran  regiment  has  400  men  in  it,  and 
another  500  men ;  how  many  more  in  the  last  than  in 
the  first  ? 

32.  3  hundred  from  6  hundred  leaves  how  many  ? 
7  hundred  less  5  hundred  is  ? 

33.  4  hundred  from  8  hundred  leaves  how  many? 
6  hundred  from  9  hundred  ?      2    hundred    from    7 
hundred  ? 


SUBTRACTION.  23 

LESSON    III. 

1.  A  horse  travelled  six  miles  in  one  hour,  and  nine 
miles  the  next ;  how  many  more  miles  did  he  travel 
in  the  second  hour  than  in  the  first  ? 

2.  A  farmer  paid  twelve  dollars  to  one  laborer,  andi 
eight  dollars  to  another;  how  much  less  did  he  pay 
the  second  laborer  than  the  first  ? 

3.  Julian  found  thirteen  eggs,  and  gave  John  seven  ; 
how  many  had  he  left? 

4.  A  hunter  killed  ten  bears  in  the  canebrake  one 
day,  and  six  on  the  next  day;  how  many  more  did  he 
kill  on  the  first  day  than  the  second  ? 

5.  6  less  6  7  less  7  ?     8  less  8  ? 

6.  7  less  6  ?  8  less  7  3    9  less  8  ? 

7.  8  less  6  \  9  less  7  \  10  less  8  ? 

8.  9  less  6  \  10  less  7  \  11  less  8  ? 

9.  10  less  6  ?  11  less  7  ?  12  less  8 1 

10.  11  less  6  ?  12  less  7  ?  13  less  8  ? 

11.  12  less  6  \  13  less  7  ?  14  less  8 1 

12.  13  less  6  ?  .14  less  7  ?  15  less  8  ? 

13.  14  less  6  ?  15  less  7  ?  16  less  8  ? 

14.  15  less  6  ?  16  less  7  2  17  less  8  2 

15.  16  less  6  ?  17  less  7  ?  18  less  8  ? 

16.  20  less  7  is  how  much  ?     30  less  7  ?   &c.,  to  90. 

17.  21  less  7  is  how  much  ?     31  less  7?  &c.,  to  91. 

18.  22  less  7  ?  32  less  7  ?  &c.,  to  92. 

19.  23  less  7  ?  33  less  7  ?  &c.,  to  93. 

20.  24  less  7  ?  34  less  7  ?  &c.,  to  94. 

21.  25  less  7  2  35  less  7  ?  &c.,  to  95. 

22.  26  less  7  ?  36  less  7  ?  &c.,  to  96. 

23.  27  less' 7  ?  37  less  7  ?  &c.,  to  97. 

24.  28  less  7  ?  38  less  7  ?  &c.,  to  98. 

25.  29  less  7  ?  39  less  7  ?  &c.,  to  99. 

NOTE. — The  pupil  should  be  required  to  go  through  several  of  the 
above  with  6  and  8  respectively. 

26.  Count  by  6's  backward  from  100  to  4. 


24  SUBTRACTION. 

27.  Count  by  7's  backwards  from  100  to  2. 

28.  Count  by  8's  backwards  from  100  to  4. 

29.  A  man  has  60  sheep  in  one  fold,  and  90  in  a 
second  fold ;   how  many  more  in  the  second  than  in 
the  first  ? 

30.  6  tens  from   9  tens  leaves  how  many  ?     7  tens 
from  10  tens  leaves  ?     8  tens  from  12  tens  ? 

31.  There   were   700   bison   in    one   herd   on    the 
prairie,  and  900  in  another  herd  ;  how  many  more  in 
the  last  herd  than  the  first  ? 

'  32.  600  from  900  leaves  how  many  ?     8  hundred 
from  11  hundred  ? 


LESSON    IV. 

1.  A  man  agreed  to  labor  for  a  farmer  15  days, 
and  left  at  the  end  of  nine  days ;  how  many  days' 
labor  did  he  still  owe  the  farmer  ? 

2.  A  hunter  on  the  Texas  prairies  killed  10  deer 
on  one  day,  and  14  the  next ;  how  many  more  did  he 
kill  on  the  second  day  than  on  the  first  ? 

3.  9  less  9  are  how  many  ?     10  less   10   are  how 
many  ? 

4.  10  less  9?  11  less  10? 

5.  11  less  9  ?  12  less  10  ? 

6.  12  less  9  ?  13  less  10  ? 

7.  13  less  9?  14  less  10? 

8.  14  less  9?  15  less  10? 

9.  15  less  9  ?  16  less  10  ? 

10.  16  less  9  ?  17  less  10  ? 

11.  17  less  9?  18  less  10? 

12.  18  less  9  ?  19  less  10  ? 

13.  19  less  9?  20  less  10? 

14.  20  less  9  are  how  many  ?     30  less  9  ?  &c.,  to  90. 

15.  21  less  9  ?  31  less  9  ?  &c.,  to  91. 

16.  22  less  9  ?  32  less  9  ?  &c.,  to  92. 

17.  23  less  9  ?  33  less  9  ?  &c.,  to  93. 


SUBTRACTION.  25 

18.  24  less  9  ?  34  less  9  ?  &c.,  to  94. 

19.  25  less  9  ?  35  less  9  ?  &c.,  to  95. 

20.  26  less  9  ?  36  less  9  ?  &c.,  to  96. 

21.  27  less  9  ?  37  less  9  ?  &c.,  to  97. 

22.  28  less  9  ?  38  less  9  ?  &c.,  to  98. 

23.  29  less  9  ?  39  less  9  ?  &c.,  to  99. 

NOTE. — The  pupil  should  go  through  in  the  same  manner  with  the 
subtraction  of  10. 

24.  Count  backwards  by  9's  from  100  to  1. 

25.  Count  backwards  by  10's  from  100  to  0. 

26.  One  squadron  of  cavalry  contains  90  mounted 
men,  and  a  second  contains  110;    how  many  more 
mounted  men  does  the  second  contain  than  the  first  ? 
9  tens  from  11  tens  leaves  how  many?     10  tens  from 
15  tens  leaves  how  many  ? 

27.  In  one  of  the  butcheries  in  Paris,  9  hundred 
sheep  were  slaughtered  in  one  day,  and  in  a  second  12 
hundred;  how  many  more  were  slaughtered  in  the 
second  than  in  the  first  ? 

28.  10  hundred   or  a  thousand  from  14  hundred 
leaves  how  many  ? 

LESSON    Y. 

1.  How  much  is  20  less  11  ? 

PROCESS. — 11  is  10  and  I.     10  from  20  leaves  10, 
and  \from  10  leaves  9. 

2.  How  much  is  45  less  11  ?     67  less  11  ?     99  less 

11  ?     88  less  11  ?     66  less  11  ? 

3.  How  much  is  25  less  12  ? 

PROCESS. — 12  is  10  and  2.     10  from  25  leaves  15, 
\ind  2  from  15  leaves  13. 

4.  How  much  is  43  less  12  ?     54  less  12  ?     60  less 

12  ?     72  less  12  ?     96  less  12  ?     84  less  12?       . 

5.  26  less  13?     52  less  13?     39  less  13?     65  less 
13? 


26  SUBTRACTION. 

6.  28  less  14  ?     42  less  14  ?     70  less  14  ? 

7.  36  less  18  ?     54  less  18  ?     90  less  18  ? 

8.  Count  backwards  by  ll's,  beginning  with  99. 

9.  Count  backwards  by  12's,  beginning  with  96. 

10.  How  much  is  30  less  20  ?     40  less  20  ?     50  less 
30?     60  less  20?     70  less  50  ?     80  less  60  ?     90  less 
30? 

11.  How  much  is  40  less  25  ? 

PROCESS. — 25  is  20  and  5.  40  less  20  is  20.  20 
less  5  is  15. 

12.  How  much  is  30  less  22  ?     50  less  36  ?     60  less 
37  ?     70  less  24  ?     80  less  48  ?     90  less  55  ? 

13.  How  much  is  55  less  20  ? 

PROCESS. — 50  less  20  is  30.  Hence  55  less  20  is 
35. 

14.  How  much  is  36  less  20  ?     48  less  20  ?     59  less 
30  2     67  less  40  ?     77  less  50  ?     87  less  30  ? 

15.  How  much  is  33  less  23  ?     66  less  46  ?     56  less 
36?     87  less  67?     97  less  57?     79  less  49?     88  less 
28? 

16.  How  much  is  55  less  22  ? 

PROCESS. — 55  is  52  and  3.  52  less  22  is  30.  30 
and  3  are  33.  Or  thus,  22  is  20  <md  2.  55  &»*  20 
i*  35.  35  less  2  w  33. 

17.  How  much  is  67  less  39  ?     78  less  23  ?     82  less 
47  2     96  less  51  ?    47  less  28  ? 

LESSON    VI. 

1.  A  boy  had  25  marbles,  and  gave  5  of  them  to 
one  comrade,  6  to  another,  and  4  to  another;  how 
many  did  he  have  left  ? 

2.  A  farmer  raised  50  bushels  of  oats  in  one  field, 
20  in  a  second,  %iid  30  bushels  in  a  third,  and  sold 
40  bushels ;  how  many  bushels  did  he  raise  on  the 
three  fields,  and  how  many  bushels  had  he  after  sell- 
ing 40  ? 


SUBTRACTION.  27 

3.  Charles'  father  gave  him  50  cents  to  make  some 
purchases.     He  bought  an  arithmetic  for  15  cents,  a 
grammar  for  10  cents,  and  a  writing-book  for  8  cents ; 
how  much  of  the  money  had  he  left? 

4.  A  traveller  had  100  miles  to  go.     On  the  first 
day  he  travelled  22  miles,  on  the  second  31  miles,  on 
the  third   17  guiles;  how  far  did  he  go  in  the  three 
days  ?     How  many  miles  had  he  yet  to  travel  ? 

5.  In  a  conservatory  there  were  20  oranges  on  one 
tree,  15  on  another,  7  on  a  third,  and  5  on  a  fourth. 
Of  these,  10  on  the  first   tree  were  ripe,    7  on  the 
second,  2  on  the  third,,  and  3  on  the  fourth ;  how 
many  oranges  were  on  all  the  trees,  how  many  were 
ripe,  and  how  many  unripe  ? 

6.  36  bees  were  sucking  honey  from  the  flowers  of 
a  wild  jessamine ;  15  of  them  flew  to  the  blossoms  of 
a  magnolia  tree,  7  to  the  roses  on  a  sweetbriar  near 
by,  while    6,  loaded  with  honey,   flew    back  to  the 
hive;  how  many  bees  still  lingered  on  the  jessamine 
vine? 

7.  A  squadron  of  cavalry  went  into  battle  one  hun- 
dred strong,  20  of  them  were  captured,  10  killed,  and 
22  wounded ;  how  many  men  were  left  in  the  squad- 
ron ? 

8.  43  and  8  and  5,  less  11,  are  how  many  ? 

9.  Twenty-eight  and  four  and  nine,  less  seven,  are 
how  many  ? 

10.  Fifty-two  and  six  and  three  and  nine,  less  two 
and  three  and  five,  are  how  many  ? 

11.  Seventy-seven  and  eleven  and  three  and  four, 
less  seven,  less  eight,  are  how  many  ? 

12.  Ninety-eight    and   two   and  eight   and   three, 
less  twenty  and  twelve,  are  how  many  ? 

13.  Eighty-six  less  twenty,  less  seven,  less  nine,  less 
twenty-two,  is  how  much  ? 


28  .  SUBTRACTION. 

LESSON    VII. 

TABLES  FOR  PRACTICE  IN  ADDITION  AND  SUBTRACTION. 

For  the  sake  of  brevity  in  these  tables,  we  will  use  the  sign  of  ad- 
dition +  ,  called  plus,  and  the  sign  of  subtraction  — ,  called  minus. 
Plus  means  more,  minus  means  less.  Thus,  7  +  6  is  read  7  plus 
6,  which  means  7  more  6,  or  7  and  6.  7  —  6  is  read  7  minus  6, 
and  means  7  less  6,  or  6  taken  from  7. 

i.? 


10  +  10 

40  +  20 

70  +  60 

80  +  80 

20  +  20 

40  +  40 

70  +  40 

90  +  30 

20  +  10 

50  +  40 

80  +  50 

70  +  80 

30  +  20 

50  +  20 

90  +  20 

90  +  90 

30  +  10 

50  +  10 

90  +  80 

60  +  50 

30  +  30 

50  +  30 

60  +  40 

60  +  20 

40  +  10 

50  +  50 

90  +  50 

30  +  80 

40  +  30 

80  +  10 

II.? 

40  +  17  40  +  34  40  +  39  40  +  41 

29  23  24  25 

38  32  31  36 

22  43  19  42 

18  21  26  28 

27  33  35  37 

III.? 

47  +  30  67  +  30  51  +  30    54  +  30 

55  49  62  64 
69  66     '  44  58 
59  57  52  63 

48  61  68  53 

56  46  65  45 

IV.? 

17  +  27    17  +  56    47  +19    49  +  28 
29        29        55        66 


SUBTRACTION.  29 


38 

+  27 

38 

+  56 

69 

+  19 

57 

+  28 

22 

22 

59 

61 

18 

19 

48 

46 

27 

32 

56 

51 

V.? 

50 

-  30 

100 

-  60 

95 

-  15 

83 

-  23 

40 

-  10 

100 

-  50 

25 

13 

70 

-  50 

110 

-  10 

55 

53 

80 

-  40 

90 

-  30 

75 

73 

70 

-  30 

80 

-  70 

65 

63 

90 

-  20 

60 

-  20 

45 

43 

VI.? 

58 

-  30 

82 

-  40 

100 

-  58 

90 

-  83 

67 

77 

47 

79 

42 

45 

76 

64 

75 

66 

67 

61 

84 

94 

38 

42 

97 

58 

24 

34 

YIL? 

93 

-  58 

93 

-  85 

36 

-  16 

89 

-  24 

47 

73 

73 

74 

76 

69 

85 

47 

67 

56 

97 

38 

38 

45 

58 

56 

24 

32 

64 

95 

8.  44  less  27  Is  how  much  ?     47  less  27  ?     44  less 
30? 

9.  If  we  add  3  to  the  number  from  which  we  are 
to  subtract  another,  what  change  does  it  make  in  the 
remainder  ? 

10.  If  we  add  it  to  the  number  which  we  subtract 
from  another  number,  what  change  do  we  make  in 
the  remainder? 


30 


MULTIPLICATION. 


11.  24  —  16  ?     24  —  23  ?     31  —  16  ?     31  -  -  23  ? 

12.  If  we    add   seven    to   the   number   subtracted 
and  also  to  the  number  from  which  it  is  taken,  is 
there  any  change  in  the  remainder  ? 

13.  5  -  3  is  how  much?     8-5?     12-9? 

14.  40  —  20  is  how  much  ?     41  —  21  ?     49  —  29  ? 

15.  One    farmer    had    32    sheep,  and  another   28 
sheep ;  each  lost  10  sheep ;    how  many  more  sheep 
did  the  first  have  than  the  second  before  the  loss  ? 
After  the  loss  ? 


CHAPTEK    III. 

t 

MULTIPLICATION. 
LESSON    I. 

1.  Five  boys  gave  each  a  dime  to  a  blind  man ; 
how  many  dimes  did  they  all  give  ?     1  and  1,  and  1 
and  1,  and  1,  are  how  many?     5  times  1  are  how 
many  ?     one  times  5  are  how  many  ones  ? 

2.  Six  boys  are  flying  kites,  and  each  boy  has  one 
kite  ;  how  many  kites  have  they  all  ?     1  and  1,  and 
1  and  1,  and  1  and  1,  are  how  many?     6  times  1  ? 
Once  6  ? 

3.  Seven  girls  are  coming  from  the  garden,  each 
with  a  bunch  of  roses ;  how  many  bunches  of  roses 
have  they  all  together  ?     1  and  1,  and  1  and  1,  and  1 
and  1,  and  1,  are  how  many  ?     7  times  1  ?     One  time 
7  is  how  many  ones  ? 

4.  Eight   boys    are    going    out   to    gather   black- 
berries, and  each   boy  has  one  basket ;    how   many 
baskets  have  they  all  \     8  times  1  are  how  many  ? 


MULTIPLICATION.  3 1 

5.  1  time    1  ? 

6.  2  times  1  ?  1  time  2  ? 

7.  3  times  1  ?  1  time  3  ? 

8.  4  times  1  ?  1  time  4  ? 

9.  5  times  1  ?  1  time  5  ? 

10.  6  times  1.?  1  time  6? 

11.  7  times  1?  1  time  7? 

12.  8  times  1?  1  time  8? 

13.  9  times  1  ?  1  time  9  ? 

14.  10  times  1  ?     1  time  10  ? 

15.  27  times  1  are  how  many?     36  times  1  ?     48 
times  1  ?     87  times  1  ?     108  times  1  ? 

16.  Twice  1  ten  is  how  many  tens  ?     3  times  1  ten  ? 
4  times  1  ten  ?     5  times  1  ten  ? 

17.  Twice  1  hundred   is    how  much  ?     6  times  1 
hundred  ?     8  times  1  hundred  ?     9  times  1  hundred  ? 

18.  Two  row   boats  have    three    oars    each  ?    how 
many  oars  have  both  boats?       3   and    3    are    how 
many  ?     Twice  3  ? 

19.  Hayes  saw  two  Esquimaux  sledges  drawn  by  8 
dogs  each  ;  how  many  do^s  drew  both  sledges  ?     8 
and  8  are  how  many  ?     Twice  8  ? 

20.  There  are  two  orange  trees  in  a  gentleman's 
garden,  each  of  which  has    20    oranges  on   it ;    how 
many  oranges  on  both  trees  ?     20  and  20  are   how 
many  ?     Twice  20  ? 

21.  A  man  bought  two  cloaks,  and  paid  9  dollars 
apiece  for  them  ;  how  much  did  he  pay  for  both  ? 

22.  How  much  is  twice    1  ? 

twice  2  ? 

twice  3  ? 

twice  4  ?  « 

twice  5  ? 

twice  6  ? 

twice  7  ? 

twice  8  ? 

twice  9? 


o*  MULTIPLICATION. 

23.  How  much  is  twice  10? 

twice  11  ? 

PROCESS.— 11  is  10  and  1.  Twice  11  is  twice  10 
and  twice  1,  or  20  and  2  (i.  e.)  22. 

24.  How  much  is  twice  12?     Twice  16?     Twice 
17  ?     Twice  18  ?     Twice  19  ? 

25.  How   much   is   twice    2   tens   or   20  ?     Twice 
30?     Twice   40?      Twice   50?     Twice   60?      Twice 
70  ?     Twice  80  ?     Twice  90  ? 

26  How  much  is  twice  1  hundred?  Twice  2 
hundred  ?  Twice  3  hundred  ?  Twice  4  hun- 
dred ?  Twice  5  hundred  ?  Twice  8  hundred  ?  Twice 
9  hundred? 

The  sign  of  multiplication  is  x ,  which  we  will  read  times.  Thus, 
2  x  12  or  3  x  6  means  2  times  12  and  3  tunes  6.  The  result  ob- 
tained by  the  multiplication  of  numbers  is  called  their  product. 


LESSON    II. 

1.  "What  do  3  oranges  at  2  cents  apiece  cost  ?     2 
+  2  +  2? 

2.  What  do  3  oranges  at  3  cents  apiece  cost  ?     3 
+  3  +  3  ? 

3.  Each  wagon  has  four  wheels  ;  how  many  wheels 
have  three  wagons  ?     How   many  wheels  have  four 
wagons  ?     How  many  have  five  wagons  ?     4   +  4  + 
4j     4  +  4  +  4  +  43     4.  +  4  +  4+4  +  4J 

4.  There  are  three  rows  of  trees,  arid  ten  trees  in 
each  row;  how  many  trees?      How  many  trees  in  4 
rows    of  ten    trees   each?     In  5  rows   of  ten   trees 
each? 

5.  3   times  10   are   how    many?     3  times  10  and 
once  10  ?     4  times  10  and  once  10  ? 

6.  3  times  1  ?  4  times  1  ?  5  times  1  ? 

7.  3  times  2  ?  4  times  2  ?  5  times  2  ? 

8.  3  times  4  ?  4  times  3  ?  5  times  3  ? 

9.  3  times  5  ?  4  times  4  ?  5  times  4  ? 


MULTIPLICATION.  33 

10.  3  times    6  ?  4  times  5  ?  5  times  5  ? 

11.  3  times    7  ?  4  times  6  ?  5  times  6 1 

12.  3  times    8  ?  4  times  7  ?  5  times  7  ? 

13.  3  times    9  ?  4  times  8  ?  5  times  8  ? 
14  3  times  10  ?  4  times  9  ?  5  times  9  ? 

15.  3  times  2  are  how  many  times  3  ?     4  times  2 
are  how  many  times  4  ? 

16.  5  times  2  are  how  many  times  5  ?     3  times  4 
are  how  many  times  3  ? 

17.  3  times  5   are  how  many  times  3  ?     4  times  5 
are  how  many  times  4  ? 

18.  How  much  is  3  times  11  ? 

PROCESS. — 11  is  10  and  1.     3  times  11  is  3  foVi&j 
10  £m£Z  3  times  1  ;  i.  e.,  30  <m<#  3,  or  33. 

19.  3  times  12?     3   times   13?     3    times   14?     3 
times  15  ?     3  times  16  ?     3  times  17  ?     3  times  18  ?     3 
times  19  ? 

20.  How  much  is  4    times  11  ?     4  times   12  ?     4 
times  13  ?     4  times  14  ?     4  times  15  ?     4  times  16  ?     4 
times  17  ?     4  times  18  ?     4  times  19  ? 

21.  How    much  is  5  times  11?     5  times    12?     5 
times  13  ?     5  times  14  ?     5  times  15  ?     5  times  16  ?     5 
times  17?     5  times  18  ?     5  times  19  ? 

22.  How  much  is  four  times  2   tens  ?     4   times  3 
tens  ?     4  tens  ?     8  tens  ?     9  tens  ? 

23.  How  much  is  3  times  2  tens  ?     3  times  3  tens  ? 
3  times  4  tens  ?     3  times  5  tens  ?     3  times  6  tens  ? 
8  tens? 

24.  How  much  is   5   times  2  tens?      3  tens?      4 
'tens  ?     5  tens  ?     6  tens  ?     7  tens  ?     8  tens  ? 

25.  How  much  is  4  times  2  hundred  ?     3  hundred? 
6  hundred  ?      5  times  3  hundred  ?      5  hundred  ?     7 
hundred  ?     3  times  2  hundred  ?     9  hundred  ?     4  hun- 
dred? 


34: 


MULTIPLICATION. 


LESSON    III. 

1.  I  planted  6  rows  of  trees  with   6  trees  in  each 
row ;  how  many  trees  did  I  plant  ?     5  times  6  and 
once  6  are  how  many  ?     7  rows  of  trees,  6  trees  in  a 
row,  are  how  many  trees  ?     8  rows  of  trees,  6  trees  in 
a  row,  are  how  many  trees  ?     6x6  and  once  6  ?     7 

x  6  and  once  6  ? 

2.  What  do  6  pounds  of  sugar  cost  at  7  cents  a 
pound  ?     at  8  cents  ?     at  10  cents  ? 

3.  There  are  7  days  in  one  week  ;  how  many  days 
in  7  weeks  ?     in  8  weeks  ?      7  and  7  and  7  and  7  and 
7  and  7  and  7  are  how  many?     7  times  7  and  once  7 
are  how  many  ? 

4.  If  flour  is  8  dollars  a  barrel,  how  much  will  5 
barrels  cost  ?     8  barrels  ?     9  barrels  ?     10  barrels  ? 

'5.  6  times  2?  7  times    2?     8  times    2? 

6.  6  times  3?  7  times    3?     8  times    3? 

7.  6  times  4  ?  7  times 

8.  6  times  5  ?  7  times 

9.  6  times  6  ?  7  times 
6  times  7  ?  7  times 
6  times  8  ?  7  times 
6  times  9  ?  7  times 


5? 
6? 
7? 
8? 
9  ? 


many 


4? 

5? 

6? 

7? 

8? 

9? 

8  times  10  ? 
times  6  ?  7  times 


8  times 
8  times 
8  times 
8  times 
8  times 
8  times 


10. 
11. 
12. 

13.  6  times  10  ?     7  times  10 

14.  6  times  5   are  how 
are  how  many  times  7  ? 
times  8  ? 

15.  6  times  7  are  how  many  times  6  ?     8  times  7 
are  how  many  times  8  ?     6  times  8  are  how  many 
times  6  ? 

6  times  11? 
19? 

7  times   11? 
19? 

}  times  11? 
19? 


16.  How  much  is 
15?     16?     17?     18? 

17.  How  much  is 
15?     16?     17?     18? 

18.  How  much  is 
15?     16?     17?     18? 


8  times  5  are  how  many 


12  ?  13  ?  14  ? 
12?  13?  14? 
12?  13?  14? 


MULTIPLICATION. 


35 


19.  How  much  is  6  times  2  tens  ?     3  tens  ?     5  tens  ? 

7  tens  ?     8  tens  ?     9  tens  ? 

20.  .How  much  is  7  times  2  tens  ?     3  tens  ?     4  tens  ? 

6  tens  ?     8  tens  ?     9  tens  ? 

21.  How  much  is  8  times  3  tens  ?     5  tens  ?     7  tens  ? 

8  tens?     9  tens?     11  tens? 

22.  How  much  is  6  times  2  hundred  ?     5  hundred  ? 

8  hundred  ? 

23.  How  much  is  7  times  3  hundred  ?     4  hundred? 

9  hundred? 

24.  How  much  is  8  times  2  hundred  ?     6  hundred  ? 

7  hundred  ? 

LESSON    IV. 

1.  An  elephant  has  two  tusks ;  how  many  tusks 
have  9  elephants?     How  many  tusks  have  10  ele- 
phants ? 

2.  In  a  certain  village  there  are  9   cottages  in  a 
row,  and  each  cottage  has  four  windows  in  front ;  how 
many  front  windows  have  all  the  cottages  ? 

3.  Ten  laborers  received  each  5  dollars  for  a  week's 
labor  ;  how  much  did  they  all  receive  together  ? 

4.  At  8  dollars  a  barrel,  how  much  will  9  barrels 
of  flour  cost  ?     Ten  barrels  ? 

2  ?  10  times 
10  times 
10  times 
10  times 
10  times 
10  times 
10  times 
10  times 


5. 

6. 

7. 

8. 

9. 
10. 
11. 
12. 

13.  9  times  10  ?     10  times  10  ? 

14.  9  times  8  are  how  many  times  9  ?     10  times  8 
are  how  many  times  10  ? 

15.  9  times  10  are  how  many  times  9  ?     10  times  2 
are  how  many  times  10  ? 


9  times 
9  times 
9  times 
9  times 
9  times 
9  times 
9  times 
9  times 


3? 

4? 
5? 
6? 
7? 
8? 
9? 


2? 
3? 

4? 
5? 
6? 
7? 
8? 
9? 


36  aiTJLTIPLICATIOIT. 

16.  9  times  2   tens  are   how   many  ?     3  tens  ?     5 
tens  i     7  tens  ?     8  tens  ?     10  tens  ? 

17.  10  times  3  tens  ?     4  tens  ?    5  tens  ?    6  tens  ?    8 
tens  ?     9  tens  ? 

18.  How  much  is  9  times  4  hundred  ?     7  hundred  ? 

8  hundred  ? 

19.  How  much  is  10  times  3  hundred  ?     6  hundred  ? 

9  hundred  ? 

20.  How  much   is   9  times  11?     12?     13?      14? 
15?     16?     17?     18?     19? 

21.  How  much  is  10  times   11?     12?     13?     14? 
15?     16?     17?     18?     19? 

22.  How  much  is  11  times  11  ? 

PROCESS. — 10  times  11  and  once  11  gives  121. 

23.  How  much  is  12  times  11  ?     12  times  12  ? 

24.  What  will  11  dozen  eggs  cost,  at  10  cents  a 
dozen  ?     At  11  cents  ?     At  12" cents  ? 

DEFINITION. — The  number  by  which  we  multiply  is  called  the  mul- 
tiplier, and  the  number  multiplied  is  called  the  multiplicand.  The 
two  are  called  factors. 


LESSON    V. 

1.  How  much  is  5  times  26  ? 

PROCESS. — 5  x  26  is  5  x  2  tens,  +  5x6,  which 
is  10  tens,  or  100,  +  30. 
ANS.— 130. 

TABLES    FOR    PRACTICE. 

IJ 


2  x  40 

5  x  90 

2  x  13 

3  x  13 

9  x  30 

.6  x  80 

2  x  23 

3  x  21 

4  x  70 

3  x  50 

2  x  14 

3  x  25 

9  x  60 

4  x  60 

2  x  42 

3  x  31 

8  x  80 

7  x  30 

2  x  35 

3  x  28 

7  x  50 

2  x  90 

2  x  49 

3  x  32 

^         MULTIPLICATION.  37 

II.? 

4  x  13  5  x  35  6  x  45  8  x  58  10  x  22 

6  x  14  6  x  58  6  x  36  8  x  47  10  x  34 

4  x  16  7  x  26  6  x  83  8  x  76  10  x  45 

5  x  22  4  x  23  7  x  42  8  x  67  10  x  56 

6  x  23  7  x  5*4  7  x  34  9  x  45  10  x  65 
4  x  42  5  x  62  7  x  53  9  x  36  10  x  72 
4  x  35  6  x  56  7  x  82^  9  x  64  10  x  91 

4.  A  man  sold  57  barrels  of  apples  at  5  dollars  a 
barrel ;  how  much  did  he  get  for  all  ? 

5.  If  he  had  sold  them  at  6  dollars  a  barrel,  how 
much  would  he  have  received?    at  7  dollars,  how 
much  ? 

6.  What  distance  does  a  man  travel  in  9  days  who 
goes  42  miles  each  day?     what  distance  in  ten  days? 

7.  Out  of  a  flock  of  sheep,  34  ewes  had  two  lambs 
each  ;  how  many  lambs  did  they  all  have  ? 

8.  In  an  orchard  there  are  8  rows  of  apple  trees, 
and  each  row  contains  22  trees ;  how  many  apple  trees, 
in  all  ? 

LESSON    VI. 

1.  How  much  is  2x2x2?     2x2x3?     2x 
2x4?      2x2x5?      2x2x6?     2x3x4? 
2x5x6? 

2.  What  are  these  products  called  ? 
ANS. —  Continued  products. 

3.  What  is  the  continued  product  of  3,  4,  and  5  ? 
5,  4,  and  3  ?     3,  5,  and  6  ?     5,  6,  and  3  ? 

4.  In  a  continued  product,  can  we  begin  the  multi- 
plication with  any  two  of  the  numbers,  without  chang- 
ing the  product  ? 

5.  What  single  number  multiplied  by  5  will  give 
the  same  product  as  2  x  3  multiplied  by  5  ?  as  2  x  2 

x5?     as  2x4x5?     as  3x4x5?     as  2x2 
x  2  x  5? 


38  MULTIPLICATION. 

6.  How  much  is  4  x5x6?     4x6x7?     3x6 
x7?     2x2x2x3x5?   2x2x2x4x4? 

7.  The  square  of  any  number  is  what  we  obtain  by 
multiplying  that  number  by  itself.      The  square  of  1 
is  1   x  1 ;  of  2,  is  2  x  2,  &c. 

8.  What  is  the  square  ofl?     21    3?     4?     5?     6? 
7?     8?     9?     10?     11?     12? 

9.  A  man  cuts  two  oranges  into  two  parts  each,  and 
then  cuts  each  of  the  parts  into  two  parts ;  into  how 
many  parts  were  both  oranges  cut  ? 

10.  A  miller  wished  to  divide  3  bushels  of  meal 
among  some  beggars.      He  divided  each    bushel  in 
four  heaps,    and  then  each   heap  in    two  parts.     In. 
how  many  parts  were  the  three  bushels  divided  ? 

11.  If  I  cut  a  melon  in    6   parts,  and  cut  each  of 
these  parts  in  5  pieces,  and  each  of  these  pieces  in  4 
parts ;  in  how  many  parts  will  the  melon  be  divided  ? 

12.  A   man  had    10   orchards ;  each   orchard  con- 
tained 10  rows  of  trees,  and  each  row  of  trees  con- 
tained 10  trees ;  how  many  trees  in  all  ? 

LESSON    VII. 

MISCELLANEOUS. 

1.  I  subtracted  15  from  a  certain  number,  and  ob- 
tained 5  for  a  remainder ;  what  is  that  number  ? 

2.  22  is   13  less  than  what  number  ?     40  is  15  less 
than  what  number  ? 

3.  What  number  is  composed  of  the  two  parts  34 
and  26? 

4.  A  farmer  sold  5  barrels  of  flour  at  5  dollars  a 
barrel,  and  10  bushels  of  wheat  at  2  dollars  a  bushel ; 
how  much  did  he  receive  for  the  flour  and  wheat  ? 

5.  The  difference  between  46  and  a  greater  number 
is  15  ;  what  is  that  number? 

6.  John  has  6  marbles,  and  William  has  3  times 
as  many,  less  10 ;  what  number  has  William  ? 


MULTIPLICATION.  39 

7.  A  farmer  sold  3  lambs  at  2  dollars  each ;  5  pigs 
at  2  dollars  each ;  8  bushels  of  wheat  at  3  dollars 
a  bushel,  and  5  barrels  of  corn  at  8  dollars  a  bar- 
rel ;  what  sum  of  money  did  he  receive  for  all  ? 

8.  July,  August,  and  October  have  each  31  days ; 
September,  March,  and  June  have  each  30  days ;  how 
many  more  5ays  in  the  first  three  than  the  last  three  ? 

PROCESS. — In  the  first  three,  there  are  3  x  31,  or  3 
x  30  -f-  3  x  1 ;  and  in  the  last  three  there  are  3  x 
30 ;  hence,  there  are  3  x  1,  or  3  more  days  in  the  first 
three  than  in  the  last. 

9.  How  much  greater  is  3  x  32  than  3  x  30  ? 
ANS.— 3  x  2. 

10.  How  much  less  is  5  x  10  than  5  x  15? 
ANS. — 5  x  5. 

11.  How  much  less  is  5  x  35  than  5  x  40  ? 
ANS. — 5  x  5. 

12.  How   much  greater  is   9     x    10  than  9x3? 
Than  9x4?     Than  9x5? 

13.  How  much  more   is    5    x    50  than  5    x    40? 
Than  5  x  41  ?     Than  5  x  45  ?     Than  5  x  49  ? 

14.  How  much  more  is  8  x  8  than  7x7? 

15.  If  a  man  buys  5  barrels  of  flour  at  0  dollars 
a  barrel,  and  sells  them  at  7  dollars,  how  much  will 
he  gain  ?     How  much  if  he  sells  at  8  dollars  a  barrel  ? 
9  dollars  a  barrel  ?     10  dollars  a  barrel  ? 

16.  Frank  has   30   chinquepins.      William    has    3 
times  as  many,  less  40,  and  Charles  has  twice  as  many, 
less  30  ;  how  many  have  they  all  together  ? 

17.  Multiply  the  sum  of  8  and  9  by  9  ?     Find  the 
sum  of  the  two  products  8  x  9and9   x  9? 

18.  Two  travellers  set  out  from  the  same  point  at 
the  same  time,  and  travel  in  opposite  directions.     One 
travels  5  miles  an  hour,    and  the  other  4  miles  an 
hour ;  how  far  are  they  apart  at  the  end  of  5  hours  ? 
At  the  end  of  6  hours'?     7  hours  ?     8  hours  ? 

19.  5x5  +  5x4:  is  how  much  ? 


4:0  DIVISION. 

20.  Two  travellers  set  out  from  the  same  point  at 
the  same  time,  and  travel  in  the  same  direction.     One 
travels  6  miles  an  hour,  and  the  other  four  miles  an 
hour;   how  far  will  the  slowest  one  be   behind   the 
other  in  5  hours  ?     6  hours  ?     9  hours  ? 

21.  5x6,  less  5  x  4,  is  how  much? 

22.  Two  travellers  set  out  at  the  same  moment,  one 
from  Richmond  and  one  from  Washington,  and  travel 
toward  each  other.     One  travels  4  miles    an    hour, 
and  the  other  5  miles  an  hour.     Putting  the  distance 
between  Richmond  and  "Washington  at "  100   miles  ; 
how  far  will  the   travellers   be  apart   3  hours'  after 
setting  out  ?     4  hours  ?     5  hours  ? 

23.  100  less  3x4  +  3x5  is  how  much  ? 

24.  Ten  times  eight,  plus  eighteen,  less  twenty,  less 
six,  are  how  many  ? 

25.  Five  times  twenty,  plus  fifty,  plus  eight,  are 
how  many  ? 

26.  Six  times  twenty,  less  five  times  ten,  are  how 
many? 

27.  If  a  man  earns  60  cents  a  day,  and  pays  out  20 
cents  a  day,  how  much  will  he  have  made  at  the  end 
of  6  days'? 


CHAPTER    IY. 

DIVISION. 
LESSON    I. 

We  have  seen  how  to  separate  large  numbers 
into  different  sets  of  smaller  ones.  Thus,  5  is  sepa- 
rated into  2  +  3,  2  +2  +  1,  2  +  1  +  1  +  1,  1  + 
1  +  1  +  1  +  1.  5  is  said  to  contain  these  smaller 
numbers.  A  number  can  be  separated  into  smaller 


DIVISION.  41 

numbers,  all  of  which  are  equal  to  each  other.  We 
say  that  the  first  number  contains  the  other  a  certain 
number  of  times. 

1.  Into  what  equal  numbers  may  every  number  be 
seperated  ? 

ANS. — Into  ones. 

2.  How  many  ones  in  6  ?     In  7  ?     8  ?     10  ?     105  ? 

3.  1  is  contained  in  6  how  many  times  ?     In  7  ? 
8?     9?     110? 

4.  Distribute  six  cannon  balls  into  heaps  of  two 
balls  each  ;  how  many  heaps  will  you  have  ? 

5.  Distribute  them  into  heaps  of  three  balls  each ; 
how  many  heaps  ? 

6.  2  is  contained  in  6  how  many  times  ? 

7.  3  is  contained  in  6  how  many  times  ?     2  -f  2  -f 
2       are  how  many? 

8.  How  many  times  can  you  take  2  from  6  and  2 
from  the  remainder  until  nothing  remains  ?     3  +  3  is 
how  much  ? 

To  divide  one  number  by  another  is  to  find  how 
often  the  latter  is  contained  in  the  former. 

9.  How  do  we  divide  12  by  2  ? 

ANS. —  We  distribute  12  into  2's,  and  then  count 
the  number  of  2's. 

10.  How  is  this  done  ? 

ANS. — Either,  1",  By  adding  2'*  thus  :  2  +  2  -f  2 
-f2  +  2  +  2,  until  we  get  12  ;  or,  2^?  by  subtracting 
'Zfrom  12  and  2  from  the  remainder,  and  so  forth, 
until  nothing  remains  •  or,  3n*,  we  may  remember, 
from  multiplication,  how  many  times  2  make  12,  or 
what  number  taken  2  times  'makes  12. 

11.  Which  of  these  is  the  shortest  process? 

The  number  which  is  divided  by  the  other  is 
called  the  dividend.  The  number  by  which  we  divide 
is  called  the  divisor.  The  result  obtained  is  called 
the  quotient. 

12.  How  many  i>air  of  spurs  will  make  eight  spurs  ? 


42  DIVISION. 

13.  How  many  pair  of  spectacles  require  ten  glasses  ? 

14.  An  ivory  hunter  gets  two  tusks  from  each  ele- 
phant which  he  kills ;  how  many  elephants  must  he 
kill  to  get  12  tusks  ? 

15.  If  Frank  studies  two  hours  *a  day,  in  how  many 
days  will  he  study  14  hours  ? 

16.  How  many  2's  in  4?      6?       8?      10?      12? 
14?     16?     18?     20? 

17.  In  what  number  is  2  contained  exactly  2  times  ? 
3  times  ?    4  times  ?     5  times  ?     6  times  ?     7  times  ?     8 

'times  ?     9  times  ?     10  times  ? 

18.  How  much  is  2  divided  by  2  ?     12  divided  by  2  ? 

4  divided  by  2  ?     14  divided  by  2  ? 

6  divided  by  2  ?     16  divided  by  2  ? 

8  divided  by  2  ?     18  divided  by  2  ? 
10  divided  by  2  ?     20  divided  by  2  ? 

The  numbers  which  contain  2  an  exact  number  of 
times  we  say  are  divisible  by  2. 

19.  How  many  pair  of  spurs  will  five  spurs  make  ? 

20.  If  we  endeavor  to  distribute  5  into  2's  what  do 
we  find  ?     7  into  2's  ?     9  into  2's  ? 

21.  "What  is  the  1  which  is  left  in  each  case  called  ? 
ANS. — The  remainder. 

22.  How  much  is  3  divided  by  2  ? 
ANS. — 1  and  1  remainder. 

23.  How  much  is  5  divided  by  2  ?     13  divided  by  2  ? 

7  divided  by  2  ?     15  divided  by  2  ? 

9  divided  by  2  ?     17  divided  by  2  ? 
11  divided  by  2  ?     19  divided  by  2  ? 

24.  "What  are  the  numbers  called  which  are  divisible 
by  2? 

ANS. — Even  numbers. 

25.  What  are  the  numbers  called  which  are  not 
divisible  by  2  ? 

ANS. — Odd  numbers. 

26.  What  remainder  results  always  when  we  divide 
the  odd  numbers  by  two  ? 


DIVISION.  43 

27.  Give  the  even  numbers  from  2  to  100. 

28.  Give  the  odd  numbers  from  1  to  99. 

29.  How  much  is  4  tens  or  40  divided  by  2  ? 
ANS. — 2  tens  or  20. 

30.  4  hundred  divided  by  2  ? 

31.  6  tens  or  60  divided  by  2  ? 

32.  6  hundred  divided  by  2  ? 

33.  8  tens   or   80   divided   by   two?      8   hundred 
divided  by  2  ? 

34.  10   tens   or   100  divided   by  2  ?     10  hundred 
divided  by  2  ? 

35.  12  tens  or  120  divided  by  2  ?     14  tens  divided 
by  2  ?     16  tens  divided  by  2  ?     18  tens  divided  by  2  ? 

36.  How  much  is  30  divided  by  2  ? 
PROCESS.— 30  is  20  -f  10.     20  divided  ly  2  is  10, 

and  10  divided  l)y  2  is  5. 
ANS. — 10  +  5  or  15. 

37.  How  much  is  50  divided  by  2  ?     70  divided  by 
2?     90  divided  by  2? 

38.  How  much  is  21  divided  by  2  ? 

PROCESS. — 21  is  20  +  1.  Hence,  21  divided  ~by  2 
t's  10  and  1  remainder. 

39.  How  much  is  23  divided  by  2  ?     24  ?     25  ?     36  ? 
45? 

40.  If  we  multiply  5  by  2,  and  divide  the  product 
by  2,  what  will  be  the  quotient  ? 

41.  If  we  divide  8  by  2,  and  multiply  the  quotient 
by  2,  what  will  be  the  product  ? 

42.  If  we  divide  any  number  by  2,  and  multiply 
the  quotient  by  2,  what  will  be  the  result  ? 

LESION    II. 

*1.  If  one  flour  barrel  holds  3  bushels,  how  many 
barrels  will  hold  9  Bushels  ?  12  bushels  ?  15 
bushels  ?  30  bushels  ? 

2.  Into  how  many  ranks  of  four  men  eaeh  can  you 


44 


DIVISION. 


divide  a  squad  of  8  soldiers  ?     a  squad  of  16  soldiers  ? 
a  company  of  30  soldiers  ?     of  40  soldiers  ? 

3.  Into  how  many  heaps  of  five  balls  each  can  you 


15?     18?     21? 


distribute  25  cannon   balls  ?     30  cannon  balls  ?  "  45 
cannon  balls  ?     50  cannon  balls  ? 

4.  How  many  3's  in  6  ?     9  ?     12  ? 
24?     27?     30? 

many   4's   in   8  ?      12?      16?      20?     24? 

36  ?     40  ? 

many  5's  in  10?      15?      25?      20?     30? 

40?'  50? 


5. 
28? 

6. 
35? 

7.  In 
Twice  ? 
times  ? 

8.  In 
Once  ? 

5  times? 


How 

32? 
How 
45? 

what  number  is  3  contained  exactly  once? 
6  times?     4  times  ?     7  times  ?     5  times  ?     9 
8  times  ?     10  times  ?     3  times  ? 
what  number  is  4  contained  exactly  twice  ? 
4  times  ?     8  times  ?      10  times  ?     6   times  ? 
7  times  ?     3  times  ?     9  times  ? 


9.  In  what  number  is  5  contained  exactly  4  times? 
Twice  ?  6  times  ?  5  times  ?  3  times  ?  7  times  ? 
9  times  ?  8  times  ?  10  times  ? 


3  divided  by  3  ? 

6  divided  by  3  ? 

9  divided  by  3  ? 
12  divided  by  3  ? 
15  divided  by  3  ? 
18  divided  by  3  ? 
21  divided  by  3? 
24  divided  by  3  ? 
27  divided  by  3  ? 
30  divided  by  3  ? 

11.  How  much  is  3  x 
many  times  ?     3  in  36  ? 

12.  How  much  is  4  x 
many  times  ?     4  in  48  ? 

13.  How  much  is  5  x 
many  times  ?     5  in  60  ? 


10.  How  much  is 
4  divided  by  4  ? 
8  divided  by  4  ? 

12  divided  by  4  ? 

16  divided  by  4? 

20  divided  by  4  ? 

24  divided  by  4  ? 

28  divided  by  4? 

32  divided  by  4  ? 

36  divided  by  4? 

40  divided  by  4  ? 
11?  3  x 


5  divided  by  5  ? 
10  divided  by  5  ? 
15  divided  by  5? 
20  divided  by  5  ? 
25  divided  by  5  ? 
30  divided  by  5? 
35  divided  by  5  ? 
40  divided  by  5  ? 
45  divided  by  5  ! 
50  divided  by  5  ? 
12?  3  in  33  how 


11?  m  4  x  12?     4  in  44  how 
11  ?     5x12?     5  in  55  how 


DIVISION.  45 

14.  3  is  contained  in  4  how  many  times,  and  what 
remainder  ?     in  5  how  many  times  ? 

15.  How  many  three  bushel  bags  can  you  fill  with 
7  bushels  of  wheat,  and  what  will  remain?     How 
many  with  8- bushels? 

16.  What  are  the  remainders  of  numbers  divided  by 
3? 

17.  Give   the   numbers   from   1  to  100  which  are 
divisible  by  3. 

18.  4  in  5  how  many  times  ?     4  in  6?     4  in  7?     4 
in  9  ?     4  in  10  ?     4  in  11  ? 

What  are  the  remainder  of  numbers  divided  by  4  ? 

19.  Give  the  numbers  from  1  to  100  which  are 
divisible  by  4. 

20.  5  in  6  how  many  times  ?     5  in  7  ?     5  in  8  ?     5 
in  9?     5  in  11?     5  in  12?     5  in  13?     5  in  14? 

What  are  the  remainders  of  numbers  divided  by  5  ? 

21.  Give  the   numbers  from  1  to   100  which  are 
divisible  by  5. 

22.  3  in  3  tens  how  many  times  ?     in  6  tens  ?     in  9 
tens?     in  12  tens?  &c. 

23.  3  in  3  hundred  how  many  times  ?     in  6  hun- 
dred?    in  9  hundred?     in  12  hundred  ? 

24.  4  in  4  tens?     in   8  tens?     in  16  tens?     in  24 
tens? 

25.  4  in  4  hundred?     in  8  hundred?     in  16  hun- 
dred? 

26.  5  in  5  tens?     10  tens?     15  tens?     in  5  hun- 
dred?    15  hundred? 

27.  Multiply  9  by  3,  4  and  5,  and  divide  the  pro- 
ducts by  3,  4  and  5  respectively. 

28.  Divide  60  by  3,  4  and  5,  and  multiply  the  quo- 
tients bv  3,  4  and  5  respectively. 

29.  A  number  multiplied  by  3   and  the  product 
divided  by  3  gives  what  result? 


DIVISION. 


LESSON    III. 

1.  If  it  requires  6  horses  to  draw  each  cannon  on 
the  field  of  battle,  how  many  cannon  will  require  12 
horses?     24   horses?      42   horses?      36 '  horses  ?      60 
horses  ? 

2.  How  many  weeks  do  14  days  make  ?     28  days  ? 
21    days?      35*  days?      49    days?      56    days?      70 
days? 

3.  If  8  leaves  only  are  left  to  grow  and  ripen  on 
each   tobacco    plant,   how    many    plants    will   yield 
16  leaves?     32  leaves?     48  leaves?     40  leaves?     80 
leaves  ? 

4.  How 
42?     54? 

5. 
42? 


many    6's   in    12  ?  24  ?     18  ?     36  ?     48  ? 
60?     30? 

many  7's  in  14?  28?      21?      35?     49? 
63  ?     70  ? 

many   8's  in  16  ?  24  ?      40  ?     32  ?     48  ? 
72  ?     80  ? 


How 
56? 

6.  How 
64?     56? 

7.  In  what  number  is  6  contained  exactly  once  ? 
times?       Twice?     5  times?     7  times  ?     8  times? 
times?     10  times? 

8.  In  what  number  is  7  contained  exactly  once? 
times  I     6  times?     3  times?     5  times?     7  times? 
times  ?     10  times  ?     8  times  ? 

9.  In  what  number  is  8  contained  exactly  once? 
times  ?     3  times  ?     4  times  ?     7  times  ?     9  times  ? 

10  times  ? 
10.  How  much  is 

7  divided  by  7  ? 
14  divided  by  7? 
21  divided  by  7? 
28  divided  by  7? 
35  divided  by  7? 
42  divided  by  7  ? 
49  divided  by  7? 
56  divided  by  7? 


times  ?     8  times  ? 

6  divided  by  6? 
24  divided  by  6  ? 
36  divided  by  6  ? 
12  divided  by  6? 
18  divided  by  6? 
48  divided  by  6? 
42  divided  by  6? 
54  divided  by  6  ? 


3 

9 

4 
9 

5 
6 


8  divided  by  8  ? 
24  divided  by  8  ? 
40  divided  by  8  ? 
32  divided  by  8? 
16  divided  by  8? 
48  divided  by  8? 
56  divided  by  8  ? 
72  divided  by  8  ? 


DIVISION.  47 

60  divided  by  6  ?     63  divided  by  7  ?     64  divided  by  8  ? 
30  divided  by  6  ?     70  divided  by  7  ?     80  divided  by  8  ? 

11.  How  much  is  6  x  11  ?     6  x  12  ?     How  much 
is  66  divided  by  6?     How  much  is  72  divided  by  6  ? 

12.  How  much  is  11  x  7  ?     12  x  7  ?     How  much 
is  77  divided  by  7  ?     84  divided  by  7? 

13.  How  much  is  11  x  8?     12  x  81     How  much 
is  88  divided  by  8?     96  divided  by  8  ? 

14.  6  in  7  how  many  times,  and  what  remainder? 
In  8?     In  9  ?     In  10  ?     In  11  ?     In  13  ?     In  14  ?     In 
15?     16?     17? 

What  are  the  remainders  of  numbers  divided  by  6  ? 

15.  7  in  15  how  many  times,  and  what  remainder  ? 
In  16?     17?     18?     19?     20?     22?     23?     24?     25? 
26?     27? 

What  are  the  remainders  of  numbers  divided  by  7? 

16.  8  in  9  how  many  times,  and  what  remainder  ? 
In  10?     11?     12?     13?     14?     19?     25?     26?     27? 
28?     29?     30?     31? 

What  are  the  remainders  of  numbers  divided  by 
8? 

17.  Give  the  numbers  from  1  to  100  which  are 
divisible  by  6. 

18.  Give  the  numbers  from  1  to  100   which    are 
divisible  by  7. 

19.  Give  the  numbers  from  1   to  100  which   are 
divisible  by  8. 

20.  6  in  6  tens  how  many  times  ?     In  12  tens  ?    In 
18  tens  ?     In  24  tens  ?     In  36  tens  ?     In  60  tens  ? 

21.  6  in  6  hundred  how  many  times?     In   12  hun- 
dred ?     In  18  hundred  ? 

22.  7  in  seven* tens  how  many  times  ?     In  14  tens? 
In  21  tens  ?     In  49  tens  ?     In  56  tens  ?     In  70  tens  ? 

23.  7  in  7  hundred?     In    14    hundred?      In   21 
hundred  ? 

24.  8  in  8  tens?     16  tens?      24  tens?      32  tens? 
56  tens  ?    80  tens  ? 


4:8  DIVISION. 

25.  8  in  8  hundred  ?     in  16  hundred  ?     in  24  hun- 
dred? 

26.  How  much  is  6  x  30   divided   by  6?     How 
much  is  56  divided  by  8  and  the  quotient  multiplied 
by  8? 

27.  6  in  40  how  many  times  ? 

PROCESS. — 40  is  36,  and  4,  6  in  36  6  times.     Hence 
6  in  40  6  times  and  remainder  4. 

28.  6  in  50  how  many  times  ?     6  in  100  ?     8  in  50  ? 
8  in  60  ?     8  in  90  ?     7  in  60  ?     7  in  80  ?     7  in  100  ? 
8  in  100  ? 

LESSON    IV. 

1.  If  I  wish  to  plant  36  trees  in  parallel  rows  of 
nine  trees  each,  how  many  rows  must  I  plant  ?     27 
trees  will  give  how  many  rows  of  9  trees  each  ?     45 
trees   how  many  rows  ?      54   trees  ?      72  trees  ?     63 
trees  ?     81  trees  ?     90  trees  ? 

2.  A  vessel  was  wrecked  on  a  rocky  coast,  and  there 
were  more  than  a  hundred  men  clinging  to  the  wreck. 
The  life-boat  from  the  shore  could  only  save  ten  men 
on  each  trip  to  the  wreck.     How  many  trips  did  its 
bold  rowers  have  to  make  in  order  to  save  30  men  ? 
40  men  ?     60  men  ?     100  men  ?     80  men  ? 

3.  How  many  9's  in  18  ?     27  ?      36  ?      45  ?     63  ? 
54?     72?     81?     90? 

4.  How  many  10's  in  20?     80?     90?     70?     60? 
50?     40?     30?     100? 

5.  What  number  contains  9  exactly  once  ?     3  times? 
Twice  ?      5   times  ?     6   times  ?     8  times  ?      7   times  ? 
4  times  ?     9  times  ?     10  times  ? 

6.  What  number  contains  10  exactly  10  times?     9 
times  ?     8  times  ?     7  times  ?     6  times  ?     5  times  ?     4 
times  ?     3  times  ?     Twice  ? 

7.  How  much  is 
9  divided  by  9  ?       10  divided  by  10  ? 


DIVISION.  49 

18  divided  by  9  ?  20  divided  by  10  ? 
27  divided  by  9  ?  30  divided  by  10  ? 
36  divided  by  9  ?  40  divided  by  10  ? 
45  divided  by  9  ?  50  divided  by  10  ? 
54  divided  by  9  ?  60  divided  by  10  ? 
63  divided  by  9  ?  TO  divided  by  10  ? 
72  divided  by  9  ?  80  divided  by  10  ? 
81  divided  by  9  ?  90  divided  by  10  ? 
90  divided  by  9  ?  100  divided  by  10  ? 

8.  How  much  is  11  x  9?     12  x  9  ?     How  many 
times  does  99  contain  9  ?     How  many  times  does  108 
contain  9? 

9.  How  much  is  11  x  10  ?     12  x  10  ?     10  in  110 
how  many  times  ?     10  in  120  how  many  times  ? 

10.  9  in  10  how  many  times,  and  what  remainder  ? 
In  11?     12?     13?     14?     15?     16?     17?     In  19? 
20?     21?     22?     23?     24?     25?     26? 

What  are  the  remainders   of  numbers  divided  by 
9? 

11.  10  in  21  how  many  times,  and  what  remainder? 
In    22?     23?     24?     25?     26?     27?     28?     29?     In 
41?     42?     43?     44?     45?     46?     47?     48?     49? 

What  are  the  remainders  of  numbers  divided  by 
10? 

12.  Give  the  numbers,  from  1  to  100,  which  are 
divisible  by  9. 

13.  Give  the  numbers,  from  1  to  200,  which  are 
divisible  by  10. 

14.  What  figure  do  all  numbers  divisible  by  10  end 
in? 

15.  9  in  9  tens  how  many  times  ?     In  18  tens  ?     27 
tens  ?     36  tens  ?     45  tens  ?     90  tens  ? 

16.  9  in  9  hundred  how  many  times  ?     18  hundred  ? 
27  hundred?     36  hundred?     45  hundred?     63  hun- 
dred? 

17.  10  in  10  tens  how  many  times?     In  20  tens? 
30  tens  ?     40  tens  ?     50  tens  ?     90  tens  ? 


50  DIVISION. 

18.  10  in  20  hundred  ?     In  3000  ?     4000  ?    60  hun- 
dred ?     In  7000  ?     In  9000  ? 

19.  9  in  86  how  many  times? 

PROCESS. — 86  is  81  and  5.    9  in  81,  9  times.    Hence, 
9  in  86,  9  times,  and  5  remainder. 

20.  9  in  97  how  many  times?     In  105?     In  111? 
113?     In  960?     963?     999?     In  456? 

21.  10  in  770  how  many  times  ?    220  ?    440  ?    1010? 
77?     78?     101?     105?     362?     456? 

22.  When  a  number  is  not  divisible  by  10,  which 
figure  of  the  number  is  always  the  remainder  ? 

23.  How  much  is  7    x    8    x  10,  divided  by  10? 
How  much  is  60  x  9,  divided  by  9  ? 

24.  How  much  is  9x8  +  3?     9x8  +  2?     9x 
7  +  5?     9x6  +  7? 

25.  How   much   is   10  x  3  +  4  ?      10  x  4  +  5  ? 
10x7  +  6?     10x8  +  9? 

LESSON    V. 

1.  A  workman,  desiring  to  finish  a  piece  of  work 
speedily,  labors  11  hours  a  day.     In  how  many  days 
will  he  have  worked  33  hours  ?     44  hours  ?     55  hours  ? 
77  hours?     22  hours?     88  hours?     99  hours?     110 
hours  ? 

2.  Twelve  eggs  make  a  dozen.     How  many  dozen 
eggs  are  24  eggs  ?     36  eggs  ?     72  eggs  ?     48  eggs  ? 
60  eggs?     84  eggs?     96eg<?s?     108  eggs  ?     120  eggs? 

3.  How  many  ll's  in  22?     33?     44?     55?     66? 
77?     88?     99?     110? 

4.  How  many  12's  in   24  ?     36  ?     48  ?     60  ?     72  ? 
84?     96?     108?     120? 

5.  What  number  contains  11  exactly  once?     Twice  ? 
4  times  ?     7  times  ?     3  times  ?     5   times  ?     9  times  ? 

6  times  ?     8  times  ?     10  times  ? 

6.  What  number  contains  12  exactly  twice  ?     Once  ? 
3  times?     5  times?     6  times?     4  times?     8  times? 

7  times  ?     9  times  J     10  times  ? 


DIVISION.  51 

• 

7.  How  much  is 

11  divided  by  11  ?  12  divided  by  12  ? 
22  divided  by  11  ?  24  divided  by  12  ? 
33  divided  by  11  ?  36  divided  by  12  2 
44  divided  by  11  ?  48  divided  by  12  ? 
55  divided  by  11  ?  60  divided  by  12  ? 
66  divided  by  11  ?  72  divided  by  12  ? 
77  divided  by  11  ?  84  divided  by  12  ? 
88  divided  by  11  ?  96  divided  by  12  ? 
99  divided  by  11  ?  108  divided  by  12  ? 
110  divided  by  11  ?  120  divided  by  12  ? 

8.  How  much  is  11  x  11  ?     12  x  11  ?     12  x  12? 

11  in  121  how  many  times  ?     11  in  132  ?  . 12  in  132  ? 

12  in  144? 

9.  11  in  12  how  many  times,  and  what  remainder  ? 
11  in  13?     in  14?     in  15  2     in  16?     in  17  ?     in  18  ? 
in  19  ?     in  20  ?     in  21  ?     in  23  ?  &c. 

What  are  the  remainders  of  numbers  divided  by 
11? 

10.  How  much  is  7  x  11  +  3?     5  x  11  +  4?     5 
x  11  +  5?     6  x  11  +  7?     6  x  11  +  10? 

11.  12  in  25  how  many  times,  and  what  remainder  ? 
in26?     27?     28?     29?     30?  -31?     32?     33?     34? 
35  ?     37  ? 

12.  How  much  is  2  x  12  +  1  ?     2  x  12  +  2?     2 
x  1.2  +  3?     2  x  12  +  4?     2  x  12  +  5?     2  x  12 
+  10?     2  x  12  +  11? 

What  are  the  remainders  of  numbers  divided  by 
12? 

13.  Give  the  numbers,  from  1  to  200,  which  are 
divisible  by  11. 

•  14.  Give  the  numbers,  from  1  to  200,  which  are 
divisible  by  12. 

15.  11  in  11  tens  how  many  times?     in  23  tens? 
33  tens  ?     66  tens  ?     99  tens  ? 

16.  11  in  11  hundred   how  many   times?     in  22 
hundred  ?     44  hundred  ?     in  88  hundred  ? 


52 


DIVISION. 


17.  12  in  12  tens  how  many  times  ?     in  24  tens? 
in  36  tens  ?     in  60  tens  ?     in  84:  tens  ?     in  96  tens  ? 

18.  12  in  12  hundred   how  many  times?     in  24 
hundred  ?     in  48  hundred  ?     in  72  hundred  ?     in  96 
hundred  ? 

19.  11  in  79  how  many  times?     in  89?     in  101? 
772?     330?     334?     67?     667?     446? 

20.  12  in  38  how  many  times?     in  360?     in  362? 
in  484  ?     in  75  ?     in  721  ?     964  ? 

21.  How  much  is  55  divided  by  11  and  the  cpo- 
tient  multiplied  by  11  ? 

22.  How  much  is  60  divided  by  12  and  the  quotient 
multiplied  by  12  ?     How  much  is  7  x  12,  divided  by 
12? 


LESSON    VI. 


1.  Pythagoras,  a  Greek  philosopher,  united  very 


DIVISION. 


53 


ingeniously  in  a  square  table  all  the  products  of  the 
numbers  from  1  to  12.  This  table  can  be  used  as  a 
table  of  quotients. 


1 

2 

3 

4 

5 
10 

G 
12 

7 

8 

9 

10 

11 

12 

2 

4 

6 

8 

14 

16 

18 

20 

22 

24 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

6 

12 

18 

24 

30 

36 

42 

48 

54 

GO 

66 

72 

7 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

83 

8 
9 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

96 

18 

27 

36 

45 

54 

G3 

72 

81 

90 

99 

108 

10 

20 

30 

40 

50 

GO 

70 

80 

90 

100 

110 

120 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

12 

24 

36 

48 

60 

72 

84 

96 

108 

120 

132 

144 

2.  The  first  row  at  the  top,  moving  from  left  to 
right,  contains  the  numbers  1,  2,  3,  &c.,  to  12.  The 
first  row  on  the  left,  moving  from  the  top  to  the  bot- 
tom of  the  table,  contains  also  the  numbers  1,  2,  3, 
&c.,  to  12. 

The  numbers  in  the  second  horizontal  row,  count- 
ing from  the  top,  and  those  in  the  second  vertical 
row,  counting  from  the  left,  are  formed  by  adding  the 

NOTE. — Horizontal  means  here  across  tho  page  from  left  to  right  or 
right  to  left  Vertical  means  up  and  down  the  page.  The  horizontal 
we  will  call  rows ;  the  vertical,  columns. 


54:  DIVISIOX. 

numbers  in  tlie  first  row  to  themselves.  Those  in  the 
third  rows  are  formed  by  adding  those  in  the  second 
rows  to  those  in  the  first  respectively.  The  numbers 
in  the  fourth  rows  by  adding  those  in  the  third  rows 
to  those  in  the  first  respectively,  &c. 

3.  The  first  row  and  the  first  column  contain  1,  2, 
3,  &c. 

The  second  row  and  second  column  contain  1,  2,  3, 
&c.,  multiplied  by  2. 

The  third  row  and  third  column  contain  1,  2,  3, 
&c.,  multiplied  by  3,  &c.,  &c. 

4.  If  we  wish  to  find  the  product  of  two  numbers, 
as  5  multiplied  by  6,  we  find  5  in  the  top  row,  and  6 
in  the  first  column  on  the  left.     We  come  down  the 
fifth  column  until  we  get  to  the  6th  row,  and  there 
find  30,  the  product. 

5.  If  we  wish  to  find  the  quotient  of  108  divided 
by  9,  we  find  9  in  the  top  row,  come  down  the  column 
to  108,  and  find  12,  the  first  number  on  the  left,  in 
the  same  horizontal  row  with  108. 

6.  How  is  the  table  of  Pythagoras  formed  ? 

7.  What  do  the  first  row  and  first  column  contain  ? 

8.  The  second  row  and  the  second  column  ? 

9.  The  fourth  row  and  the  fourth  column  ? 

10.  Write  the  table  on  your  slate. 

11.  Use  the  table  to  count  by  2's  from  2  to  24. 
Count  by  3's  from  3  to  36.     Count  by  5's  from  5  to 
60.     Count  by  9's  from  9  to  108.     Count  by  ll's  from 
11  to  132.     Count  by  12's  from  12  to  144. 

12.  Begin  with  the  numbers  24,  36,  48,  &c.,  in  the 
column  on  the  right,  and  count  backward  by  2's,  3's, 
4's,  &c.,  respectively. 

LESSON    VII. 

1.  Seven  are  how  many  times  2  ?  How  many  times 
3?  4? 


DIVISION.  55 

2.  Ten  are  how  many  times  2  ?     3  ?     4  ?     5  ? 

3.  Nine  are  how  many  times  2?     3  ?     4  ?     5  ? 

4.  Eleven  are  how  many  times  3?     4?     6  ? 

5.  Twelve   are  how  many  times  4  ?     6  ?     3  ?     2  ? 
5?     8? 

6.  Thirteen  are  how  many  times  2  ?     3  ?     4  ?     5  ? 
6?     7?     8?     9? 

7.  Fourteen  are  how  many  times  2  ?     7  ?     5  ?     3  ? 
4? 

8.  Fifteen  are  how  many  times  3?     5?     6?     4? 

9.  Sixteen  are  how  many  times  4  ?     5  ?     6  ?     3  ? 

10.  Seventeen  are  how  many  times  3  ?     4  ?     5  ?     2  ? 
6?     9  ? 

11.  Eighteen  are  how  many  times  2?   3?   4?    6?  9? 

12.  IN  in  eteen  are  how  many  times  6?  3?   10?  2?   5? 

13.  Twenty  are  how  many  times  2  ?     3  ?     4  ?     5  ? 
6?     7?     8?     9?     10? 

14.  Twenty-one  are  how  many  times  7  ?     3  ?     8  ? 
2  ?     4  ?  "  6  ?     10  2 

15.  Twenty-two  are  how  many  times  2  ?     3  ?     4  ? 
5?     6?     7?     8?     9?     10?     112 

1G.  Twenty-three  are  how  many  times  2  ?     3  ?     4  ? 
6?     8?     10?     11? 

17.  Twenty-four  are  how  many  times  2  ?     3  ?     4  ? 
5?     6?     6?     8?     9? 

18.  Twenty-five  are  how  many  times  2?     3?     4? 
5?     6?     7?     8?     10? 

19.  Twenty-six  are  how  many  times  3  ?     4  ?     5  ? 
6?     7?     8?     9?     10? 

20.  Twenty-seven  are  how  many  times  3  ?     4  ?     5  ? 
6?     7?     8?     10?     11? 

21.  Twenty-eight  are  how  many  times  3  ?     4  ?     5  ? 
6  ?     8  ?     10  ? 

22.  Twenty-nine  are  how  many  times  3  ?     4  ?     5  ? 
6?     7?     8?     9?     10? 

23.  Thirty  are  how  many  times  3  ?     4  ?     5  ?     C  ? 
7?     8?     9?     10? 


56  DIVISION. 

24.  Thirty-one  are  how  many  times  3  ?     4  ?     5  ? 
6?     7?     8?     9?     11? 

25.  Thirty-two  are  how  many  times  9  ?     10  ?     8  ? 
7?     6?     5?     4?     3? 

26.  Thirty- three  are  how  many  times  6  ?     8  ?     7  ? 
4?     9?     5*     10?     3? 

27.  Thirty-four  are  how  many  times  7  ?     3  ?     9  ? 
10?     6?     8?     4?     5? 

28.  Thirty-five  are  how  many  times  3?     4?     5? 
6?     7?     83     9?     10? 

29.  Thirty-six  are  how  many  times  3  ?     4  ?     5  ?     6  ? 
7?     8?     9?     11? 

30.  Thirty-seven  are  how  many  times  3  ?     4  ?     5  ? 
7?     8?     9?     10? 

31.  Thirty-eight  are  how  many  times  2?     4?     3? 
5?     6?     7?     8?     9? 

32.  Thirty-nine   are  how  many  times  3  ?     4  ?     5  ? 
6?     7?     8?     9?     10?     11? 

33.  Forty  are  how  many  times  8  ?     10  ?     6  ?     4  ? 
3?     9?     5?     7? 

34.  Forty-one  are  how  many  times  4  ?     5  ?     6  ?     7  ? 
8?     9?     10? 

35.  Forty-two  are  how  many  times  4  ?    5  ?     6  ?    7  ? 
8?     9?     10?     11? 

36.  Forty-three  are  how  many  times  4  ?     5  ?     6  ? 

7  ?     8  ?     9  ? 

37.  Forty-four  are  how  many  times  4  ?      5  ?     6  ? 
7?     8?     9?     11? 

38.  Forty-five  are  how  many  times  4  ?     5  ?     6  ?     7  ? 

8  ?     9  ?     10  ? 

39.  Forty-six  are  how  many  times  4  ?     5  ?     6  ?     7  ? 
8?     9?     10? 

40.  Forty-seven  are  how  many  times  4  ?     5  ?     6  ? 
7?     8?     9?     10? 

41.  Forty-eight  are  how  many  times  4?     5?     6? 
7?     8?     9?     10?     12? 

42.  Forty-nine  are  how  many  times  4  ?    5  ?    6  ?  &c. 


DIVISION.  57 

43.  Fifty  are  how  many  times  4  ?     5?     6?     7?     8? 
9?     10?     11? 

LESSON    VIII. 

1.  Fifty-one  are  how  manv  times  10?     5  ?     3  ?     G? 
9?     7?     8?    4? 

2.  Fifty-two  are  how  many  times  4  ?     8  ?     7  ?     6  ? 
12?     10?     13? 

3.  Fifty-three  are  how  many  times  9?     10?     6? 
7?     8?     12?     11? 

4.  Fifty-four 'are  how  manv  times  9?     10?     6?     7? 
8?     12?     11? 

5.  Fifty-five  are  how  many  times  4  ?     8?     10?     12? 

11  ?     2  ? 

6.  Fifty-ei^ht  are  how  many  times  2  ?     3  ?     4?     5  2 

6  ?     7  ?     8  ?     9  ?     10  ? 

7.  Sixty  are  how  many  times  3?     4?     12?     15? 
30? 

.8.  Sixty-four  are  how  many  times  4?    8?    16?    12? 
30? 

9.  Sixty-five  are  how  many  times  5  ?     6  ?     7  ?     8  ? 
10?     12?     13? 

10.  Sixty-nine   are  how  many  times  6  ?     7  ?     10  ? 
20? 

11.  Sixty -eight  are  how  many  times  11?     12?     8? 

7  ?     10  ?     6  ? 

12.  Sixty-seven  are  how  many  times  8  ?     7  ?     11  ? 

12  ?     5  ?     9  ? 

13.  Seventy  are  how  many  times  2  ?     6  ?     7  ?     8  ? 
^  ?     11 ?     10  ? 

14.  Seventy -two  are  how  many  times  3  ?     5  ?     7  ? 
6?     8?     9?     12? 

15.  Seventy-three  are  how  many  times  6  ?     7  ?     8  ? 
9?     10? 

16.  Seven tv-six  are  how  many  times  3  ? .  4  ?     5  ? 
6?     7?     8?  "9?     10?     11? 

3* 


58  DIVISION. 

17.  Seventy-seven  are  how  many  times  6?     7  ?     8  ? 
9?     10?     11? 

18.  Eighty  are  how  many  times  7  ?     8  ?     9  ?     10  ? 
11?     20?     40? 

19.  Eighty-one  are  -how  many  times    7?     8?     9? 
10?     11?     123     6? 

20.  Eighty-four  are  how  many  times  6  ?     7  ?     8  ? 
9?     10?     11?     12? 

21.  Eighty-five  are  how  many  times  6  ?     7  ?     9*? 
9?     tOf    111     12? 


22.  Eighty-eight  are  how  many  times  7  ?     8  ?     9 
10?     11?     12? 

23.  Eighty-nine  are  how  many  times  7  ?     11  ?     12? 
8?     9?     101 

24.  Ninety  are  how  many  times  10  ?     12  ?     7?     9  ? 
8  ?     6  ?     30  ? 

25.  Ninety-one  are   how  many  times   7?     8?     9? 
10?     11?     12?     30? 

26.  Ninety-four  are  how  many  times  12  ?     7  ?     10  ? 
9?     8?     11? 

27.  Ninety-live  are  how  many  times  9  ?     10  ?     7  ? 
8  ?     11  ?     12  ? 

28.  Ninety-six    are   how  many   times    9  ?     7  ?     8  ? 
10?     11?     12? 

29.  Ninety-seven  are  how  many  times  S?     9?     10? 
11?     12? 

30.  Ninety-eight  are  how  many  times  8  ?     9  ?     10  ? 
11  ?     12  ?     30  ? 

31.  Ninety-nine  are  how  many  times  8  ?     9?     10  ? 

32.  One  hundred  are  how  many  times  4  ?     5  ?     20  ? 
50?     9?     10?     2?     8?     25?. 

33.  One  hundred  and  eight  are  how  many  times  9  ? 
12?     3?     6?     25?     20? 

34.  One  hundred  and  ten  are  how  many  times  10  ? 
11?     5?     2?     20? 

35.  One  hundred  and  twenty  are  how  many  times 
10?     12?'    6?     20?     5?     8? 


MISCELLANEOUS.  «v  59 

36.  One  hundred  and  forty-four  are  how  many 
times  12?  9?  6?  20?  50? 

LESSON    IX. 

1.  Divide  126,  180,  162,  and  138  each  by  2. 

PROCESS. — 126  is  12  tens  and  6  units.  12  tens  con- 
tain 2  6  tens,  or  60,  times,  and  6  contains  2  3  times. 
Hence,  126  contains  2  63  times. 

2.  Divide  114,  150,  168,  243,  and  108  each  by  3.     • 
PROCESS. — 114  is  11  tens  and  4  units,  or  9  tens  and 

2  tens  and  4  units.     3  in  9  tens,  3  tens,  or  30  times. 

3  m  24,  8  times.     Hence,  3  in  114,  38  times. 

3.  Divide  140,  180,  408,  252,  and  156  each  by  4. 

4.  Divide  120,  240,  315,  225,  and  550  each  by  5. 

5.  Divide  96,  126,  162,  186,  198,  and  216  each  by 
6. 

6.  Divide  91,  105,  154,  224.  and  308  each  by  7. 

7.  Divide  104,  176,  200,  216,  and  404  each  by  8. 

8.  Divide  126,  207,  333,  306,  and  405  each  by  9. 

9.  Divide  500,  690,  780,  320,  and  450  each  by  10. 

10.  Divide  154,  198,  253,  374,  and  495  each  by  11. 

11.  Divide  192,  300,  480,  492,  and  708  each  by  12. 


, 


CHAPTER    Y. 

MISCELLANEOUS. 
LESSON    I. 


'or  the  sake  of  abbreviation,  the  sign  _•-  is  used  in  division  to  in- 
dicate "  divided  by."  30  JL.  6  is  read,  30  divided  by  (j.  This  sign,  as 
well  as  the  x  for  " times,"  or  "multiplied  by,"  will  be  used  hereafter, 
when  it  is  more  convenient  to  do  so. 

1.  Thomas  divided  12  apples   equally  among  his 


60  *  MISCELLANEOUS. 

three  brothers ;  how  many  apples  did  he  give  each 
of  them  ? 

2.  If  4  yards  of  cloth  cost  20  dollars,  how  much 
does  one  yard  cost  ?     2  yards  ? 

3.  Mary  divided  10  roses  equally  among  her  5  play- 
mates, how  many  did  she  give  to  each  one  ? 

4.  If  I  pay  35  cents  for  5  pounds  of  sugar,  how 
much  does  one  pound  cost  ? 

5.  42  bushels  of  corn  meal  were  divided  equally 
among  Y  persons;  how  many  bushels  did  each  receive  ? 

6.  60  dollars  are  paid  for  12  cords  of  wood ;  how 
much  does  the  wood  cost  per  cord '? 

Y.  If  10  oranges  are  worth  as  much  as  40  apples, 
how  many  apples  are  worth  as  much  as  one  orange  ? 

8.  If  63  men  reap  a  field  in  one  day,  how  many 
men,  laboring  at  the  same  rate,  will  reap  it  in  9  days  ? 

9.  How  many  barrels  of  flour  can  be  bought  for  84 
dollars,  at  12  dollars  a  barrel  ?     at  Y  dollars  a  barrel? 
at  14  dollars  a  barrel  ? 

10.  Two  boats  on  the  Mississippi  are  100  miles 
apart.     The  hindmost  boat  gains  on  the  other  5  miles 
an  hour.     In  how  many  hours  will  it  overtake  the 
other  ? 

11.  A  man,  having  86  dollars,  bought  12  cords  of 
wood,  and  had  2  dollars  left ;  how  much  did  he  give 
for  each  cord  ? 

12.  What  is  the  price  of  a  pound  of  beef,  if  12 
pounds  can  be  bought  for  60  cents  ?     for  84  cents  ? 
96  cents  ? 

13.  One    man   can    do    a   piece    of    work   in    25 
days  ;   how  long  will  it  take  5  men  to  do  the  same 
work  ? 

14.  5  men  can  do  a  piece  of  work  in  10  days ;  ho^jv 
long  will  it  take  one  man  to  do  the  same  work  1 

KOTE. — Each  day's  labor  of  each  man  is  considered  the  same  in 
these  examples. 


MISCELLANEOUS.  61 

15.  If  one  pipe  discharges  a  cistern  of  water  in  84 
hours,  how  long  will  it  take  7  pipes  of  the  same  size 
to  discharge  the  cistern  ? 

16.  If  12  pipes  discharge  a  cistern  in  96  hours,  how 
long  will  it  take  one  pipe  of  the  same  dimensions  to 
discharge  it  ? 

17.  Two  boats  are  50  miles  apart,  approaching  each 
other.     One  is  moving  at  the  rate  of  6  miles  an  hour, 
and  the  other  4  miles  an  hour ;  how  long  before  they 
will  come  together  ? 

18.  For  65  dollars,  how  many  sheep  can  you  buy, 
at  6  dollars  apiece,  and  how  much  money  would  you 
have  left  ? 

19.  If  51  dollars  be  divided  among  3  men,  how 
many  dollars  will  each  receive  ? 

20.  If  12  tons  of  coal  cost  96  dollars,  how  much 
does  one  ton  cost  ?     2  tons  ? 

21.  In  how  many  days  will  5  men  reap  a  field  of 
wheat,  if  40  men  reap  it  in  one  day  ? 

22.  A  laborer  engaged  to  work  4  months  for  96 
dollars;  how  much c did  he  receive  for  each  month? 
How  much  for  each  week,  allowing  4  weeks  to  each 
month  ?     How  much  each  day,  allowing  6  working 
days  to  each  week  ? 

23.  Six  whaling  vessels  brought  from  their  voyages 
1800  barrels  of  sperm  oil,  and  each  vessel  brought  the 
same  number  of  barrels  ;  how  many  barrels  did  each 
bring  ? 

24.  How  much  butter  at  12  cents  a  pound  can  be 
bought  with  6  dozen  eggs   at  10  cents  a  dozen?      6 

les  10  are  how  many  times  12  ? 
.  4  times  9  are  how  many  times  3  ? 

26.  4  times  10  are  how  many  times  5  ?     8  ? 

27.  5  times  12  are  how  many  times  15  ?     6  ? 
6  times  7  are  how  many  times  14  ? 
How  many  times  12  are  9  times  4? 

30.  A  man  travels  8  miles  an  hour  for  9  hours, 


62  MISCELLANEOUS. 

how  many  hours  must  a  man  travel  6  miles  an  hour 
in  order  to  go  the  same  distance  ? 

31.  How  many  times  7  are  3  times  14  ? 

32.  How  many  times  12  are  9  times  4  +  24:  ? 

33.  5  times  9  +  9  are  how  many  times  6  ? 

34.  7  times  12  —  14  are  how  many  times  5  ? 

35.  How  many  yards  of  cloth   at  12  dollars  a  yard 
can  be  bought  for  360  dollars  ? 

36.  How  many  tons  of  coal  at  11  dollars  a  ton  can 
be  bought  for  136  dollars,  and  how  many  dollars  will 
remain  ? 

37.  8  times  8  are  how  many  times  4  ?     32  ?     16  ? 

38.  7  times  10  are  how  many  times  5  ?     7  ?     14  ? 

39.  How  long  will  it  take  a  man  to  ride  72  miles, 
if  he  rides  8  miles  an  hour  ?    9  miles  an  hour  ?    3  miles 
an  hour  ?     6  miles  an  hour  ?     4  miles  an  hour  ? 

40.  10  times  6  are  how  many  times  12  ?     20  ?     5  ? 
4  ?     30  ? 

41.  How  much  is  67  less  9  times  3  ?     less  4  times 
9? 

42.  A  boat  sailed  up  stream   8  miles  an  hour  for 
12  hours,  and  came  back  over  the  same  distance  in  6 
hours  ;  how  many   miles  an  hour  did  she  sail  down 
stream  ? 

43.  How  much  is  25  less  45  -=-  5  ? 

44.  How  much  is  27  ~  3  and  the  quotient  multi- 
plied by  4  't 

LESSON    II. 


1.  If  3  oranges  cost  9  cents,  how  much  will  5 
oranges  cost  ? 

PROCESS. — 1.  If  3  oranges  cost  9  'cents,  1  orange 
costs  3  cents.  2.  If  1  orange  costs  3  cents,  5  oranges 
cost  15  cents. 


MISCELLANEOUS.  63 

2.  If  4  lemons  cost  24  cents,  how  much  will  7  lem- 
ons cost  ? 

3.  If  12  barrels  of  apples   cost  108.  dollars,  how 
much   will  one   barrel  cost?     If  1  barrel  of  apples 
costs    9,  dollars,  how  much  will  3  barrels   cost '{     4 
barrels  ? 

4.  A  man  travels  16  miles  in  4  hours ;  how  far  does 
he  travel  in  1  hour  ?     How  far  in  6  hours  ?     7  hours  ? 
9  hours  ? 

5.  How  many  workmen  can  do  in   1   day  what  4 
workmen  can  do  in  12  days  ? 

ANS. — 12  times  as  many,  or  48  workmen. 

6.  How  many  workmen  will  require  12  days  to  do 
what  48  workmen  can  do  in  1  day  ? 

ANS. — 48  is  12  times  as  many  as  the  number  of 
workmen  required. 

Hence  48  divided  by  12,  or  4  workmen. 

7.  3  workmen  in  15  days  will  do  as  much  as  15 
workmen  in  how  many  days? 

PROCESS. — 1.  The  work  done  by  3  workman  in  15  days 
'can  be  done  in  one  day  by  15  x  3,  or  45  workmen. 
2.  What  45  workmen  can  do  in  one  day,  15  workmen 
can  do  in  as  many  days  as  45  -j-  15,  or  3  days. 

8.  How  many  workmen  will  do  in  4  days  what  2 
workmen  require  28  days  to  accomplish  ? 

PROCESS. — 1.  The  work  which  2  workmen  can  do  in 
28  days,  can  be  done  l>y  one  workman  in  2  x  28,  or 
56  days.  2.  What  one  workman  can  do  in  56  days, 
can  be  done  in  4  days  by  as  many  workmen  as  56  ~ 
4,  6>^  14  workmen. 

9.  15  laborers  in  4  days  do  as  much  as  5  laborers 
in days  ? 

10.  24  laborers  in  2  days  do  as  much  as  16  laborers 
in days  ? 

11.  12  laborers  in  6  days  do  as  much  as  9  laborers 
in days? 

12.  9  laborers  in  10  days  do  as  much  as  30  laborers 
in days  3 


64  MISCELLANEOUS. 

13.  10  laborers  in  18  days  do  as  much  as  60  laborers 
in days  ? 

14.  10  laborers  in  12  days  do  as  much  as  20  labor- 
ers in days  ? 

15.  12  laborers  in  12  days  do  as  much  as  18  laborers 
in days  ? 

16.  40  laborers  in  3  days  do  as  much  as  6  laborers 
in days  ? 

17.  15  days'  work  of   3  men  is  equal  to  9  days' 
work  of men  ? 

18.  15  days'  work  of  4  men  is  equal  to   5   days' 
work  of men  ? 

19.  3  days'  work  of  10  men  is   equal  to   16   days' 
work  of men  ? 

20.  6  days'  work  of  6  men   is   equal  to  12  days' 
work  of men  ? 

21.  5  days'  work  of  15  men  is  equal  to  15  days' 
work  of men  ? 

22.  24  days'  work  of  5  men  is  equal  to  12  days' 
wrork  of men  ? 

23.  24  days'  work  of  4  men  is  equal  to  48  days' 
w^ork  of men  ? 

24.  How  much  will  30  pounds  of  sugar  cost  if  6 
pounds  cost  42  cents  ? 

25.  If  10  pencils  cost   30  cents,  what  will  be  the 
cost  of  7  pencils  ?     of  9  ?     of  15  ?     of  12  ? 

26.  If  7  days'  board  cost  14  dollars,  what  will  be 
the  cost  of  30  days'  board  ? 

27.  If  8  quarts  of  milk  cost  48   cents,  how  much 
will  30  quarts  cost? 

28.  A  merchant  bought  20  yards  of  cloth  for  100 
dollars ;  he  gave  10  yards  of  it  for  some  flour  which 
cost  10  dollars  a  barrel ;  how  many  barrels  did  he 
get? 

29.  A  greyhound,  chasing  a  rabbit,  is  30  yards  be- 
hind the  rabbit,  and  gains  on  him  6  yards  every  3 
minutes ;  in  how  many  minutes  will  he  overtake  him  ? 


MISCELLANEOUS.  65 

30.  A  man  hired  a  laborer  and  agreed  to  give  him 
10  dollars  for  every  3  days'  work ;  how  much  did  he 
have  to  pay  him  each  week  of  6  working  days,  and 
how  much  for  each  month  of  4  weeks  ? 

31.  If  6  oranges  are  worth  1  pine-apple,  and  2  pine- 
apples are  worth  1  melon,  how  many  oranges  are  worth 
4  melons  ? 

32.  If  9  bushels  of  apples  cost  18  dollars,  how  much 
do  100  bushels  cost? 

33.  If  60  pounds  of  pork  cost  300  cents,  how  much 
will  9  pounds  cost  ?  10  pounds  ?     3  pounds  ? 

34.  If  a  man  travels  56  miles  in  8  hours,  how  far 
does  he  travel  in  3  hours  ?     4  hours ?     5  hours? 

35.  On  the  12th  of  November,  25  meteors  appear 
and  shoot  across  the  sky  every  2  hours ;  how  many  ap- 
pear in  3  hours  ?     In  5  hours  ? 

36.  A  certain  number  of  rations  last  20  soldiers  5 
days ;  how  long  would  they  last  2  soldiers  ?     50  sol- 
diers?    25  soldiers? 

37.  40  days'  rations  for  5  men  are  8  days'  rations 
for  — —  men  ? 

38.  16  days'  rations  for  5  men  are  10  days'  rations 
for men  ? 

39.  18  days'  rations  for  6  men  are  12  days'  rations 
for men  ? 

40.  30  days'  rations  for  60  men  are  18  days'  rations 
for men  ? 

41.  3  days'  rations  for  90  men  are  10  days'  rations 
for men  ? 

42.  24  days'  rations  for  10  men  are  3  days'  rations 
for men  ? 

43.  16  days'  rations  for  6  men  are  12  days'  rations 
for men  ? 

44.  8  days'  rations  for  8  men  are  2  days'  rations 

•  men  ? 

5.  10  yards  of  cloth  at  8  dollars  a  yard  will  pay 
how  many  barrels  of  flour  at  5  dollars  a  barrel  ? 


s 


66  MISCELLANEOUS. 


46.  9  times  8  are  how  many  times  24  -=-  2  ? 

47.  9  times  11  less  18  are  how  many  times  36  — 

48.  10  times  6  and  12  are  how  many  times  40  —  8  ? 

49.  12  times  7  and  15  are  how  many  times  44  —  4? 

50.  4  times  12  and  8  are  how  many  times  49  —  7  ? 

51.  Bought  20  barrels  of  flour,  at  6  dollars  a  barrel, 
and  6  yards  of  cloth,  at  5  dollars  a  yard.     Paid  for 
the  whole  with  15  loads  of  hay.     How  many  dollars 
did  I  get  for  each  load  of  hay  ? 

52.  If  4  cords  of  wood  are  worth  24  dollars,  and 
10  cords  of  wood  are  given  for  40  thousand  feet  of 
plank,  how  many  dollars  will  each  thousand  feet  of 
plank  cost  ? 

53.  5  men  bought  a  horse  for  63  dollars,  and  paid 
two  dollars  a  week  for  keeping  him ;  at  the  end  of  8 
weeks,  they  sold  him  for  54  dollars.     How  much  did 
each  man  lose  by  the  bargain  ? 

54.  A  man  bought  12  bales  of  cotton  at  5  hundred 
dollars  for  every  two  bales.     He  sold  all  the  cotton  at 
6  hundred  dollars  for  every  three  bales.     How  many 
hundred  dollars  did  he  lose  ? 

55.  What  is  the  rent  of  12  acres  of  land,  that  of 
16  acres  being  20  dollars  ? 

56.  How  long  will  10  months'  rations  for  300  sol- 
diers, issued  at  the  same  rate,  last  a  garrison  of  500 
soldiers  \ 


MULTIPLES    AND   DIVISORS.  67 


CHAPTER    VI. 

MULTIPLES    AND     DIVISORS. 
LESSON    I. 

1.  12  contains  6  an  exact  number  of  times.     By 
adding  6  to  itself  once,  or  multiplying  it  by  2,  we 
obtain  12.     Hence  12  is  called  a  multiple  of  6.    Thus, 
24  is  a  multiple  of  6,  and  50  is  a  multiple  of  5. 

2.  When  is  one  number  said  to  be  a  multiple  of 
another  number  ? 

ANS. —  When  one  number  contains  another  number 
an  exact  nuinber  of  times,  it  is  said  to  be  a  multiple 
of  that  number. 

3.  6  divides  12  exactly,  and  hence  is  called  a  divisor 
of  12,  or  one  of  its  factors.     In  the  same  manner,  4 
is  &  factor  or  divisor  of  12.     5  is  one  of  the  factors  or 
divisors  of  60. 

4.  What  is  meant  by  a  factor  or  divisor  of  a  given 
number  ? 

ANS. — Any  number  which  is  contained  an  exact 
nujnber  of  times  in  the  given  number. 

5.  Every  number  contains  itself  once.    5  contains  5 
once,  100  contains  100  once.     Hence,  every  number 
is  a  multiple  and  divisor  of  itself. 

6.  What   number    divides    exactly   all    numbers? 
How  often  is  1  contained  in  5  ?     In  6  ?     In  500  ? 

7.  What  two  divisors  has  every  number,  at  least  ? 
ANS. — The  number  itself,  and  one. 

8.  Give  4  numbers  which  are  multiples  of  9. 

9.  Give  5  numbers  which  are  multiples  of  10.     Of 
11.     Of  12. 


68  MULTIPLES    AND    DIVISORS. 

10.  Give  3  multiples  of  each  of  the  numbers,  20,  30, 
and  40. 

11.  Find  some  numbers  which  are  factors  or  divisors 
of  15.     Of  20.     Of  40.     Of  50. 

Including  one  and  the  numbers  themselves,  respec- 
tively— 

12.  Find  the  3  divisors  of  4.    Of  9.    Of  25.    Of  49. 

13.  What  are  the  four  divisors  of  8  ?     of  10  ?     of 
14?      15?      21?      22?      26?      33?     34?     35?     38? 
39?      46?      51?      55?      57?     58?      62?      65?     69? 
74?      77?     82?      85?     86?     87?     91?      93?     94? 
95? 

14.  What  are  the  five  divisors  of  16  ?     of  81  ? 

15.  What  are  the  six  divisors  of   12  ?      18  ?     20  ? 
28?      32?      44?      45?      50?      52?     63?     68?     75? 
76?     92?     98?     99? 

16.  What  are  the  seven  divisors  of  64? 

17.  What  are  the  eight  divisors  of  24?     of  30? 
40  ?     42  ?     54  ?     56  ?     66  2     70  ?     78  ?     88  ? 

18.  What  are  the  nine  divisors  of  36  ?     of  100  ? 

19.  What  are  the  ten  divisors  of  48  ?     of  80  ? 

20.  What  are  the  12  divisors  of  60  ?     of  72  ?     of 
84  ?     of  90  ?     of  96  ? 

LESSON    II. 

A  number  which  has  several  divisors  is  called  a 
common  multiple  of  these  divisors.  Thus,  10  is  a 
common  multiple  of  2  and  5,  because  2  and  5  are 
both  contained  in  10  an  exact  number  of  times.  4  is 
a  common  multiple  of  2  and  4.  12  is  a  common 
multiple  of  1,  2,  3,  4,  6,  and  12. 

1.  What  is  a  common  multiple   of  two  or  more 
numbers  ? 

ANS. — It  is  any  number  which  contains  each  of  the 
given  numbers  an  exact  number  of  times. 

2.  Find  some  common  multiples  of  5  and  4.    Of  6, 
3,  and  2.     Of  3  and  4. 


MULTIPLES    AND    DIVISORS. 


69 


3.  What  is  the  least  common  multiple  of  two  or 
more  numbers  ? 

ANS. — It  is  the  smallest  number  which  these  num- 
bers  will  exactly  divide. 

4.  What  is  the  least  common  multiple  of  2  and  3  ? 
of  3  and  4  ?    of  4  and  5  ?    of  5  and  10  ? 

5.  of  2  and  4  ?    of  3  and  5  ?    of  5  and  6  ?    of  5  and 
100? 

6.  of  2  and  5'?    of  3  and  6  ?    of  4  and  7  ?    of  6  and 
0? 

7.  of  2  and  6  ?    of  3  and  7  ?    of  4  and  8  ?    of  7  and 
1? 

8.  of  2  and  7?    of  3  and  8  \    of  4  and  9  ?    of  8  and 

9.  of  2  and  8  ?     of  3  and  9  ?    of  4  and  10  ?     of  9 
id  30  ? 

10.  of  2  and  9  ?     of  3  and  10  ?    of  4  and  11  ?    of 
and  8  ? 

11.  What  is  the  least  common  multiple  of  2,  3,  and 

PROCESS. — The  least  common  multiple  of  2  and  3 
1  6,  of  6  and  4  is  12.     Hence ,  12  is  the  least  common 
\dtiple  of  2,  3,  and  4. 

12.  What  is  the  least  common  multiple 

of  2,  3,  and  5  ?  of  3,  4,  and  5  ? 
of  2,  3,  and  6  ?  of  3,  4,  and  6  ? 
of  2,  3,  and  7  ?  of  3,  4,  and  7  ? 

13.  What  is  the  least  common  multiple  of  2,  3,  4, 
id  6? 

PROCESS. — The  least  common  multiple  of  2  and  3 
•  6,  of  6  and  4  is  24,  ^  24  #;i6?  6  is  24.  .SWtf,  ^ 
ast  common  multiple  of  2,  3,  4,  6  is  24. 

14.  What  is  the  least  common  multiple 

of  2,  3,  4,  5  ?  of  3,  4,  5,  6  ?  of  4,  5,  6,  8  ? 
of  2,  3,  4,  7  ?  of  3,  4,  5,  10  ?  of  4,  5,  6,  9  ? 
of  2,  3,  4,  10  ?  of  3,  4,  5,  8  ?  of  4,  5,  6,  10  ? 

15.  What  is  the  least  common  multiple  of  2,  3,  4, 


70  MULTIPLES    AND    DIVISORS. 

LESSON    III. 

1.  Give  two  or  more  numbers  of  which  3  is  a 
divisor.     3  is  called   a  common  divisor  or  common 
factor  of  these  numbers. 

2.  What  numbers  are  common  divisors  of  8  and 
12? 

ANS. — 2  and  4 — because  they  divide  8  and  12  with- 
out  a  remainder. 

3.  What  numbers  are  common  divisors   of  6,  12, 
and  24  ? 

ANS. — 2,  3,  and  6. 

4.  When  is  a  number  said  to  be  a  common  divisor 
or  common  factor  of  two  or  more  numbers  ? 

ANS. —  When  it  is  contained  an  exact  number  of 
times  in  each  of  these  numbers. 

5.  What  number  is  a  common  divisor  of  all  num- 
bers ? 

6.  Which  is  the  greatest   common  divisor  of  8  and 
12? 

ANS. — 4.  Because  it  is  the  greatest  number  which 
divides  8  and  12  without  a  remainder. 

7.  What  is  the  greatest  common  divisor  of  6,  12, 
and  24  ? 

ANS.— 6. 

8.  What  is  the  greatest  common  divisor  of  two  or 
more  numbers  ? 

ANS. — The  greatest  number  which  is  contained  ex- 
actly in  these  numbers. 

9.  What  is  the  greatest  common  divisor  of  15  and 
20?     of  6  and  24  ?     of  12  and  30  ?     of  25  and  45  ? 
of  22  and  33  ? 

10.  What  is  the  greatest  common  divisor  of  4,  8, 
and  16? 

PROCESS. — The  greatest  common  divisor  of  8  and 
16  is  8 ;  of  8  and  4  -is  4.  Hence ,  4  is  the  greatest 
common  divisor  of  4,  8,  and  16. 


MULTIPLES   \XD    DIVISORS.  71 

11.  What  is  the  greatest  common  divisor  of  5,  10, 
and  15  (     of 

12.  What  is  the  srreatest  common  divisor  of  IS  and 
of  '24:  and  3'  0  and  54:  i     of  24:  and  100* 


LESSOR    IV. 

A  nnmber  which  has  no  other  divisors  except  the 
number  1  and  the  number  itself  is  called  a  prime 
number.  Thus,  2  has  no  divisor  except  itself  and  1 ; 
3  has  no  divisor  except  3  and  1 ;  5  has  no  divisor  ex- 
cept 5  and  1.  tire.  Hence,  2,  3.  and  5  are  prime  num- 
bers, 1  is  also  a  prime  number. 

1.  Are  any  of  the  even  numbers  which  are  greater 
than  2,  prime  numbers  •     Why  not  \ 

2.  Find  the  prime  numbers  from  the  number  1  to 
the  number 

3.  Find  the  prime  numbers  between  25  and  50. 
4:.  Find  the  prime  numbers  between  50  and 

5.  Find  the  prime  numbers  between  75  and 
The  numbers  which  are  not  prime  are  the  prod:: 

of  two  or  more  numbers: — I  >  _        _          >  _   x  3. 

_        4:,  or  2  x  2  x  2.      i-i  >-   x  3,  or  2  x  ± 
3,  or  2  x  2  x  2  x  3. 

The  divisors  or  fac:  .\re  called  the//r/ffi£ 

fact  2.  2,  2,  3.  the /•/•///!<:-  factors  of  L'4r. 

6.  2i  is  the  continued  product  of  what  4:  numK 

9  is  the  product  of  what  2  prime  numbe  il      -  is  the 
continued  product  of  what  three  prime  fact-;   - 

7.  What  are  the  prime  factors  of  any  number  ? 
Axs. — The  smallest  numbers  whose  continued  pro- 
duct id II  given  number. 

^    Find  the  2  prime  factors  of  each  of  the  number- 
,  U,  15,  21, 

I":.  "".  --.  B&,  ^      87, 
(Thirty-four  numbt 
9.  Find  3  prime  f  :'  each  of  the  nuniK 


72  FRACTIONS. 

12, 18,  20,  27,  28,  30,  42,  44,  45,  50,  52,  63,  66,  68,  70, 
75,  76,  78,  92,  98,  99.     (Twenty-two  numbers.) 

10.  What  are  the  4  prime  factors  of  16,  24,  36,  40, 
54,  56,  60,  81,  84,  88,  90, 100  ?     (Twelve  numbers.) 

11.  What  are  the  5  prime  factors  of  32,  48,  72,  80  ? 
(Four  numbers.) 

12.  Find  the  6  prime  factors  of  each   of  the  num- 
bers, 64  and  96.      (Two  numbers.) 

13.  There  are  32  +  22  +  12  +  4+2    numbers, 
from  1  to  100,  which  have  factors ;   how  many  are 
there  ? 

14.  How  many  prime  numbers  from  1  to  100  ? 

15.  What  number  multiplied  by    itself   gives  4? 
25?     9?     49?     16?     36?      64?      81-?     100?     121? 
144  ?     169  ?     196  ? 


CHAPTER    VII. 

FRACTIONS. 

LESSON    I. 
Halves,  Thirds,  and  Fourths,  or  Quarters. 


1.  IF  a  line  be  divided  into  two  equal  parts,  what  is 
each  one  of  these  parts  called  ? 

ANS. — One-half  of  the  line. 

2.  How  do  you  get  the  half  of  an  apple  or  orange, 
or  any  thing  ? 

ANS. — Divide  it  into  two  equal  parts  and  take  one 
of  these  parts. 

3.  How  many  half  cents  in  1  cent  ?     How  many 
half  cents  in  two  cents  ?     How  many  half  apples  in 


FRACTIONS.  73 

three  apples?     How  many  half  cents  in  50  cents? 
How  many  half  dollars  in  8  dollars  ? 

4.  How  many  halves  in  1  ?     2  ?     3  ?     4  ?     5  ?     6  ? 
7  ?     8  ?     9  ?     10  ?     61 ?     40  ?     500  ? 

5.  How  do  you  find  the  number  of  halves  in  any 
number  ? 

ANS. — Multiply  the  number  by  2. 

6.  How   many   half  apples   can    you   make    of  2 
half    apples?     How    many    cents    are    equal    to    10 
half  cents?     How  many  dollars  do  12  half  dollars 
make? 

7.  How  many  ones  in  2  halves  ?     4  halves  ?      8 
halves?      6   halves      10   halves?      12   halves?      24 
halves  ?     48  halves  ?     100  halves  ? 

18.  How  do  you  get  the  number  of  ones  or  units  in 
y  number  of  halves  ? 
ANS. — Divide  the  number  of  halves  by  2. 

9.  How  many  half  apples  in  3  apples  and  one  half 
apple  ?     How  many  dollars  in  5  dollars  and  a  half 
dollar  ?     How  many  half  yards  of  cloth  in  7  yards 
and  a  half  yard  ? 

10.  How  many  halves  in  10  and  one-half?     11  and 
one-half?     15  and  one-half?     50  and  one-half?     60 
and  one-half?     4  and  one-half?     19  and  one-half? 

11.  How  many  apples   in  3  half  apples?      How 
many  dollars  in  7  half  dollars  ?     What  remains  over 
in  each  case? 

12.  How  many  ones  in  7  halves?     9  halves?     11 
halves?      13   halves?      27   halves?     99   halves?     57 
halves  ?  45  halves  ?     23  halves  ?     31  halves  ? 

t!3.  7  divided  by  2  is  how  much? 
ANS. — 3  and  one-half. 
14.  2  in  27  how  many  times  ?     2  in  99  ?     2  in  31  ? 
n  23  ?     2  in  81  ?     2  in  11  ? 
15.  1  half  of  2  apples  is  -    -  ?     one-half  of  3  dol- 
's  is  how  much  ? 
16.  How  much  is  one-half  of  2  ?     of  3  ?     of  5  ?     of 


74  FRACTIONS. 

6?     of  7?     of  8?     of  9?      of  10?     of  11?     of  12? 
of  25?     of  100?     50?     40?     32?     22?     400?     61? 

17.  How  do  we  get  the  half  of  a  number  ? 
ANS. — Divide  it  by  2. 

18.  3  is  the  half  of  what  number?     6  is  the  half 
of  what   number  ?     the  half  of  what  number  is   7  ? 
9?     12?     20?    ^30?     25?     60?     15?     19? 

PROCESS. — 3  is  the  half  of  two  3's,  or  2  x  3.     7  is 
the  half  of  2  x  7,  &c.,  &c. 

19.  How  much  is  three  halves  of  12  ? 

PROCESS. — One-half  of  12  is  6  ;  hence,  three  halves 
of  12  is  three  times  6. 

20.  How  much  is  three  halves  of  24  ?     of  50  ?     of 
30?     of  18?     of  20? 

21.  How  much  is  7  halves  of  20  ?     18  ?     30  ?     12  ? 

22.  How  much  is  9  halves  of  18  ?     24  ?     10  ?     90  ? 

23.  How  much  is  11  halves  of  12  ?    14  ?    30  ?    22  3 


24.  If  a  line  be  divided  into  three  equal  parts,  what 
is  each  part  called  ? 

ANS. — One-third. 

25.  How  do  you  get  the  third  of  an  apple,  orange, 
or  of  any  thing  ? 

26.  How  do  you  get  two-thirds  of  any  thing  ? 

27.  How  many  thirds  of  an  apple  in  one  apple  ? 
in  2  apples  ?     How  many  thirds  of  a  cent  in  3  cents  ? 
How  many  thirds  of  a  yard  of  cloth  in  6  yards  of 
cloth  ? 

28.  How  many  thirds  inl?     2?     3?     4?     5?     8? 
7?     6?     9?     10?     30?     40?     12?     11?     300? 

29.  How  do  you  find  the  number  of  thirds  in  any 
number  ? 

30.  How  many  apples  do  3  thirds  of  an  apple  make  ? 
How  many  dollars  are  equal   to  6  thirds  of  a  dollar? 
How  many  oranges  do  12  thirds  of  an  orange  make  ? 


FRACTIONS.  75 

31.  How   many  ones   in   3   thirds  ?     6  thirds  ?     9 
thirds?      15  thirds?      12   thirds?      18   thirds?      24 
thirds  ?   21  thirds  ?    27  thirds  ?    99  thirds  ?    60  thirds  ? 

32.  How  do  you  find  the  number  of  ones  or  units 
in  any  number  of  thirds  ? 

33.  How  many  thirds  of  an  orange  in  3  oranges 
and  1  third  of  an  orange  ?     in  3  oranges  and  2  thirds  ? 
How  many  thirds  of  a  bushel  of  meal  in  6  bushels 
and  one-third  of  a  bushel  ?     How  many  thirds  of  a 
dollar  in  8  dollars  and  2  thirds  of  a  dollar  ? 

34.  How  many  thirds  in  3  and  one-third  ?    in  3  and 
2  thirds  ?     in  5  and  2  thirds  ?     in  30  and  a  third  ? 
21  and  2  thirds  ?     in  40  and  one-third  ?     in  11  and  2 
thirds  ?     in  100  and  one-third  ? 

35.  How  many  oranges  can  you  make  of  4  thirds  of 
an  orange  ?     How  many  dollars  do  8  thirds  of  a  dollar 
make?    How  many  thirds  remain  in  each  case?    How 
many  bushels  of  meal  do  10  thirds  of  a  bushel  make  ? 
What  remains  ? 

36.  How  many  ones  in  4  thirds  ?      5  thirds  ?      7 
thirds?    8  thirds?    11  thirds?    13  thirds?    16  thirds? 
20  thirds  ?     61  thirds  ?     100  thirds  ? 

37.  7  divided  by  3  is  how  much  ? 
ANS. — 2  and  one-third. 

38.  3  in  13  how  many  times?     in  14?     80?  62? 
44?     22?     8? 

39.  How  much  is  one-third  of  2  ?     Show  by  the 
picture  above  that  one-third  of  two  is  2  thirds  of  one. 
How  much  is  one-third  of  3  ?     4  ?     5  ?     6?     7  ?  8  ? 
9?      10?      11?      12?      24?      27?     15?      18?  30? 
60?     90?     600?     900?     99?     22?     17?     31? 

40.  How  do  you  find  one-third  of  any  number  ? 
How  much  is  2  thirds  of  6  ? 

PROCESS. — One-third  of  §  is  2.     Hence,  2  thirds  of 
6  is  2  x  2,  or  4. 

41.  How  much  is  two-thirds  of  21  ?     2  thirds  of  9  ? 
2  thirds  of  5  ?     of  7  ?     of  100  ?     of  60  ? 


76  FRACTIONS. 

42.  How  much  is  4  thirds  of  6  ? 

PROCESS. — One-third  of  6  is  2.     Hence,  4  thirds  of 
6  is  4:  x  2,  or  8. 

43.  Which  is  the  greater,  4  thirds  of  a  number  or 
the  number  itself?     How  much  is  3  thirds  of  6  ? 

44.  How  much  is  4  thirds  of  21  ?     24  ?     27  ?     60  ? 
15?  10?     12? 

45.  How  much  is  5  thirds  of  5  ?     12  ?     21  ?     24  ? 
27?     60?     33? 

46.  How  much  is  6  thirds   of  27  ?     9  ?     12  ?     18  ? 
24?     30?     42?     36? 

47.  How  much  is  10  thirds  of  15  ?     12  ?     21  ?     27  ? 
20?     60?     90?     99? 

48.  2  is  one-third  of  what  number  ?     6  is  one-third 
of  what  number  ? 

49.  One-third  of  what  number  is  7  ?     9  ?     12  ?     8  ? 
20?     30?     60?     19?     15?     100?    300?     600? 

PROCESS. — 2  is  one-third  of  three  2's,  or  3  times  2. 
6  is  one-third  of  3  times  6,  dec. 


50.  If  a  line  be  divided  into  4  equal  parts,  what  is 
each  one  of  these  parts  called  ?      • 

ANS. — One-fourth  of  the  line,  or  sometimes  one- 
quarter  of  the  line. 

51.  How  do  you  get  one-fourth  or  a  quarter  of  an 
apple  or  orange,  or  of  any  thing  ?     How  do  you  get  2 
fourths  or  quarters  ?     3  fourths  or  quarters  ? 

ANS. — Divide  it  into  four  equal  parts,  and  take  one 
of  the  parts,  two  of  the  parts,  and  three  of  the  parts, 
respectively. 

52.  How  many  fourths  of  an  apple  in  one  apple  ? 
How  many  fourths  of  a  bushel  of  meal  in  2  bushels  of 
meal  ?    How  many  fourths  of  a  pint  of  wine  in  one  pint 
of  wine  ?     How  many  quarter  dollars  in  5  dollars  ? 

53.  How  many  fourths  in  1  ?     2  ?     3  ?    4  ?    5  ?     6  ? 


FRACTIONS.  77 

7?     8?     9?     10?     11?      12?     13?     15?     20?     30? 
40?     100?     200?     202? 

54.  How  do  you  find  the  number  of  fourths  in  any 
Dumber  ? 

55.  How  many  fourths  in  3  and  one-fourth  ? 
PROCESS. — There   are  12  fourths  in  3,  and  these, 

added  to  one-fourth  give  13  fourths. 

56.  How  many  fourths  in  6  and  2  fourths  ?     7  and 
3  fourths  ?     20  and  one-fourth  ?     How  many  quarter 
dollars  in   10  dollars  and  3  quarter  dollars?     How 
many  fourths  in  30  and  3  fourths  ?     in   100  and  2 
fourths  ?     in  12  and  3  quarters  ? 

5  7.  How  many  oranges  do  4  fourths  of  an  orange 
make  ?  How  many  dollars  do  12  quarter  dollars 
make  ?  How  many  gallons  of  wine  do  eight  quarter 
gallons  make  ? 

58.  How  many  ones  in  4  fourths ?     8  fourths?     12 
fourths  ?     24  fourths  ?     32  fourths  ?     28  fourths  ?     60 
fourths  ?     40  fourths  ?     400  fourths  ?     88  fourths  ? 

59.  How  do  you  find  the  number  of  ones  in  any 
number  of  fourths  ? 

60.  How  many  oranges  do  5  quarter  oranges  make  ? 
How  much  over? 

61.  How  many  dollars  do  6  quarter  dollars  make? 
How  much  over  ? 

62.  How  many  yards  of  cloth  do  17  quarter  yards 
make  ?     How  much  over  ? 

63.  How  many  ones  in  5  fourths?     6  fourths?     7 
fourths  ?     21  fourths  ?     22  fourths  ?     61  fourths  ?     49 
fourths  ?     37  fourths  ?     39  fourths  ? 

64.  If  we  divide  9  apples  among  four  boys,  how 
many  whole  apples  would  each  boy  get  ?     In  order  to 
give  each  boy  an  equal  share,  what  must  we  do  with 
the  remaining  apple  ?     9  divided  by  4  is  how  much  ?    , 

ANS. — 2  and  one  fourth. 

65.  If  we  divide  10  apples  equally  among  4  boys, 
what  would  be  the  share  of  each  one  ? 


78  FRACTIONS. 

ANS. — 2  apples  and  2  fourths  of  an  apple. 

66.  If  we  divide  11  apples  equally  among  4  boys, 
what  would  be  the  share  of  each  ? 

67.  4  in  13  how  many  times?     in  14  ?     15?     22? 
21?     25?     27?     30?     62?      63?     43?     103?     101? 
401? 

68.  How  much  is  one-fourth  of  2  ?     of  3  ?     Show 
by  the  picture  of  lines  above  that  one-fourth  of  2  is 
2  fourths  of  1.     Show  that  one-fourth  of  3  is  3  fourths 
of  1. 

69.  How  much   is   one-fourth  of  2  ?     3  ?     4  ?     5  ? 
6?     7?     8?     9?      10?      11?      12?     13?     16?     18? 
20?     44?     60?     64?     80?     400?     444? 

70.  How  do  you -find  one- fourth  of  any  number  ? 

71.  How  much  is  2  fourths  of  8  ?     3  fourths  of  8  ? 
4  fourths  of  8  ?     5  fourths  of  8  ?     10  fourths  of  8  ? 

72.  5  fourths  of  eight  is  how  many  times  8  ? 
ANS. — Once  8  and  one-fourth  of  8. 

73.  How  much  is  3  fourths  of  12  ?     of  9  ?     of  24  ? 
of  28?     32?     80?     100?     400?      How   much   is    2 
fourths  of  8  ?     16  ?     12  ?     40  ?     100  ?     400  ? 

74.  How  much  is  5  fourths  of  56  ?     60  ?     92  ?     80  ? 
20?     28?     36?     40?     48? 

75.  How  much  is  7  fourths  of  12  ?     96  ?     20  ?     16  ? 
24?     52?     40?     44? 

76.  How  much  is  10  fourths  of-  8  ?     12  ?     24  ?     6  ? 
16?     80?     400? 

77.  2   is   one-fourth  of  what   number  ?     3  is  one- 
fourth  of  what  number?     One-fourth  of  what  num- 
beris4?    5?    6?     7?     8?     9?     10?     11?    12?    60? 
40?     100?     200?     15?     400?     900? 

78.  8  is  -what  part  of   16  ?     24  ?     32  ?     6  is  what 
part  of  24  ?     12  ?     18  ?     7  is  what  part  of    21  ?  28  ? 
14?    10  is  what  part  of  40?    20?    30?    60  is  what  part 
of  120  ?     240  ?      180  ?     9  is  what  part  of  27  ?     36  ? 
18  ?     12  is  what  part  of  36  ?     15  is  what  part  of  60  ? 
14  is  what  part  of  28  ?     100  is  what  part  of  400  ? 


FRACTIONS.  79 

79.  3  halves  is  3  -f-  ?     5  halves  is  5  -4-  ?     4  thirds 
is  4  +  i     5  fourths  is  5  -f-  ?     100  fourths  is  100  H-  ? 

We  use  figures  to  write  briefly  halves,  thirds,  and 
fourths,  or  in  general,  fractions.  (The  sign  —  signifies 
equal  to.)  3  halves  is  written  f  ;  5  fourths  is  written 
£  ;  20  thirds  is  written  2^-. 

80.  What  does  the  number  above  the  line  si^nd 
for? 

ANS. — It  gives  the  number  of  halves,  fourths,  or 
thirds. 

81.  What  does  the  number  below  the  line  show  ? 
ANS. — To  write  halves  we  put  2  below  the  line  /   to 

write  thirds  we  put  3  ;    to  write  fourths  we  put  4 
below  the  line. 

82.  Give  the  figures  and  their  places  which   will 
represent  7  halves,  10  thirds,  14  fourths.     Read  the 

frflf»tinnq  -6-°-    4_4    2-7     3.    5. 

&     ,2    )      35      45453' 

83.  5  and  one-half,  0  and  2  thirds,  8  and  3  fourths, 
&c.,  are  written  without  any  sign  between  the  frac- 
tion and  the  whole  number,  thus  :  5|-,  6f,   8f,  and 
these  are  called  mixed  numbers. 

Write  100  and  one-fourth,  20  and  2  fourths,  5  and 
2  thirds,  7  and  a  half.  Read  25|,  42i,  8j. 

NOTE. — In  the  chapters  preceding  Chapter  VII.  the  word  number 
means  whole  number.  In  this  and  succeeding  chapters  it  is  applied 
to  fractions  as  well  as  whole  numbers. 


80  FRACTIONS. 

LESSON     II. 


1.  If  a  line  be  divided  in  five  equal  parts,  what  do 
we  call  one  part  of  it  ? 

ANS. — One-fifth  of  the  line. 

2.  If  a  line  be  divided  into  6  equal  parts  ?     Seven 
equal  parts  ? 

ANS. — One  part  is  called  one- sixth  and  one-seventh, 
respectively. 

3.  How  do  you  get  one  seventh  of  anything?    one- 
sixth  ?      one-fifth  ?      2   sevenths  ?      3    sevenths  ?      4 
sevenths  ?    5  sevenths  ?     2  fifths  ?    3  fifths  ?    4  fifths  ? 
3  sixths?     5  sixths? 

4.  If  we  conceive  the  number  1   divided  into  five 
equal  parts,  what  will  these  parts  be  called  ?     If  it  is 
divided  into  six  equal  parts,  what  do  we  call  them  ? 
into  7  equal  parts  ? 

5.  How  many  fifths  inl?     2?     3?     4?     5?     6? 
7?     8?     9?     10?     11?      12?      20?     15?     40?     60? 
24?     32?     100?     200? 

6.  How  many  fifths  in  2?     3?     4?     5?     6?     7? 
8?     9?     12?     10?     11?     20?     15?     40?     30?    31? 
50  ?     100  ?     400  ? 

7.  How  many  sevenths  inl?    2?    3?    4?     5?    6? 
7?     8?     9?     10?     11?     12?     75?     16?     19?     20? 
30?     41?     100?     120? 

8.  How  do  you  find  the  number  of  fifths  in  any  num- 
ber ?   the  number  of  sixths  ?    the  number  of  sevenths  ? 

9.  How  many  apples  do  5  fifths  of  an  apple  make  ? 
6  fifths  of  an  apple  ?     How  many  dollars  are  equal  to 
12  sixths  of  a  dollar?     to  14  sevenths  of  a  dollar  ? 

10.  How  many  ones  in  5  fifths  ?    6  fifths  ?    7  fifths  ? 
8  fifths  ?     9  fifths  ?     10  fifths  ?    15  fifths  ?    20  fifths  ? 


FRACTIONS.  81 

25  fifths  ?    26  fifths  ?    2T  fifths  ?    28  fifths  ?    29  fifths  ? 
30  fifths  ?     60  fifths  ?     100  fifths  ?     200  fifths  ? 

11.  How   many   ones   in   6   sixths  ?     7  sixths  ?     8 
sixths  ?    9  sixths  ?     10  sixths  ?     11  sixths  ?     12  sixths  2 
21  sixths?     30   sixths?    36   sixths?     48   sixths?     60 
sixths?      72   sixths?      100    sixths?      31   sixths?      32 
sixths  ?     34  sixths  ?     in     600  sixths  ? 

12.  How  many  ones  in  7  sevenths  ?    8  sevenths  ?     9 
sevenths?     10 sevenths?      11  sevenths?     12  sevenths? 
13  sevenths  ?    14  sevenths  ?  15  sevenths?   21  sevenths  ? 
42  sevenths  ?  35  sevenths  ?    70  sevenths  ?  84  sevenths  ? 
in  100  sevenths?    490  sevenths  ?     210  sevenths  ? 

13.  How  do  you  find  the  number  of  ones  in  any 
number  of  fifths  ?     in  any  number  of  sevenths  ?     in 
any  number  of  sixths  ? 

14.  7  in  8  how  many  times?  in  10  ?  in  26  ?  in  30  ? 

15.  6  in  13  how  many  times  ?     in  14  ?     in  15  ?     in 
28  ?     in  29  ? 

16.  5  in  11  how  many  times  ?     in  12  ?     in  13  ?     in 
24  ?     in  39  ? 

17.  If  I  wish  to  divide  13  oranges  among  5  boys, 
how    many    oranges    and    what   parts    of  an    orange 
would  the  equal  share  of  each  boy  be  ?     What  would 
be  the  equal  share  of  each  of  6  boys  ?   of  each  of  7  boys  ? 

18.  How  many  fifths  in  3   and  one-fifth?     20  and 
3  fifths  ?    60  and  4  fifths?    5  and  one-sixth  how  many 
sixths  y§4  and  3  sixths  ?     100  and  5  sixths?     8  and  2 
sevenths  how  many  sevenths  ?     10   and  3  sevenths  ? 
1 5  and  5  sevenths  ?     40  and  6  sevenths  ? 

19.  How  much  is  one-fifth  of  2  ?     Draw  lines  on 
3ur  slate,  and  show  that  one-fifth  of  2  is  2  fifths  of 
ne.      Show  that   one-sixth  of  5    is   5   sixths  of  1. 
liow  that  one-seventh  of  3  is  equal  to  3  sevenths 
'1. 

20.  How  much  is  one-fifth  of   3  ?     4?     5?     10? 
>?     20?     24?     25?      30?      31?      35?      37?      40? 
I?     50.?    100?     200?     500?     150?     250? 


82  FRACTIONS. 

21.  How  much  is  one-sixth  of  2  ?    3  ?     4  ?     5  ?     6  ? 
12?     13?      14?     18?      24?      28?      29?     30?      36? 
58?     72?     70?     100?     600?     120?     300? 

22.  How  much  is  one-seventh  of  2  ?     3  ?     4  ?     5  ? 
6?    7?    11?     12?     13?     14?     28?     30?     35?    36? 
42?     49?     70?     63?     100?     84?     490?     700? 

23.  How  much  is  2  fifths  of  10  ?     3  fifths  of  45  ?  4 
fifths  of  50?  6  fifths  of  30?  4  fifths  of  80?  5  fifths  of  25? 

24.  How  much  is  3  sixths  of  12  ?     4  sixths  of  18  ? 
5  sixths  of  96  ?     6  sixths  of  72  ?     7  sixths  of  42. 

25.  How  much  is  one-seventh  of  84?    2  sevenths  of 
84  ?     3  sevenths  of  84  ?    4  sevenths  of  21  ?    5  sevenths 
of    28  ?     6  sevenths  of  35  ?     7  sevenths  of  98  ?     10 
sevenths  of  70  ? 

26.  3  is  one-fifth  of  what  number  ?     One-fifth  of 
what  number  is  4  ?    5?     6?     7?     8?     9?     10?     12? 
60?     100?     25?     33?     200? 

27.  3  is  one-sixth  of  what  number  ?     One-sixth  of 
what  number  is  2  ?     4  ?     5  ?     6  ?     7?     8?     9  ?     10  ? 
11?     12?      13?      14?      15?      20?      22?      30?     40? 
50?     100?     45?     200?     400? 

28.  2   is    one-seventh    of    what     number?      One- 
seventh  of  what  number  is  3?     4?     5?     6?     7?     8? 
9?     10?     11?     12?     13?     14?     20?    21?    30?    33? 
40?     100?     200?     300? 

29.  What  part  of  15  is  3?     What  part  of  30  is 
5?     6?     What  part  of  42  is  6?     7?     WhaP^rt  of 
35  is  5  ?     7  ?     What  part  of  72  is  12  ?     What  part  of 
63  is  9  ?     of  80  is  16  ?     8  is  what  part  of  16  ?     24  ? 
32?     40?     48?     56?     1  is  what  part  of  2?     3?     4? 
5  ?     6  ?     7  ?     100  is  what  part  of  200  ?     300  ?     400  ? 
500  ?     600  ?     700  ? 

30.  4  fifths  of  1  is  one-fifth  of  4,  or  4  divided  by 
what  ?    6  sevenths  is  one-seventh  of  6,  or  6  divided  by 
what  I     5  sixths  is  one-sixth  of  5,  or  5  divided  by  what  \ 

31.  4  fifths  is  written  f  •   6  sevenths  is  written  ^  ; 
4  sixths  is  written  |.     What  does  the  number  under 


FRACTIONS.  83 

the  line  show  ?    What  does  the  number  over  the  line  ? 
What   figures,   and  how  placed,  stand   for  1  sixth? 

2  fifths?     3    sevenths?     4   sevenths?     3    sixths?     1 
fifth  ?     1    seventh  ?     4   and   2   sevenths  ?     5    and  5 
sixths?     100  and  3  fifths? 

32.  Read  the  following  fractions:  J,  f,  JT%  ^S  Y-? 
V,  f,   I,  J,  -4*,  47i,  3i,  5f,  20i. 

LESSON   III. 

1.  If  a  line  be  divided  into  8  equal  parts,  what  is 
each  one  of  those  parts  called  ?     ANS. — One-eighth  of 
the  line.     If  the  line  is  divided  into  nine  equal  parts, 
what  is  each  part  ?     ANS. — One-ninth.     If  the  line  be 
divided  into  10  equal  parts,  what  is  each  part?     ANS. 
— One-tenth  of  the  line. 

2.  How  do  you  find  the  eighth,  ninth,  and  tenth  of 
any  thing?     How  do  you  get  4  eighths  of  a  thing? 

3  ninths  ?     7  tenths  ? 

3.  How  many  eighths  of  a  dollar  in  one  dollar? 
in   2   dollars  ?     4  dollars  ?     How  many  ninths  of  a 
gallon  of  wine  in  one  gallon  ?    3  gallons  ?    How  many 
tenths  of  a  dollar  in  5  dollars? 

4.  How  many  eighths  inl?    2?    3?    4?    5?     6? 
7?    8?    9?    10?    11?    12?    20?    30?    40?    22?    35? 
44  ?     100  ?     200  ? 

5.  HW  many  ninths  inl?     2?     3?    4?     5?     6?     7? 
8?     9?     10?     11?     12?     13?     14?     15?     16?     20? 
24?     30?     40?     100?     120?     300?     900? 

6.  How  many  tenths  inl?    2?    3?    4?    5?    6?    7? 
8?     9?     10?     11?     12?     13?     14?     15?     16?     20? 
37?     49?     56?     100?     240?     500?     1000? 

7.  How  many  bushels  do  16   eighths  of  a  bushel 
make  ?     How  many  dollars  are  equal  to  21  dimes  or 
tenths  of  a  dollar  ?     How  many  yards  of  cloth  are 

Iual  to  27  ninths  of  a  yard  ? 
8.  How  many  ones  in  8  eighths  ?     10  eighths  ?     12 


84: 


FRACTIONS. 


eighths  ?  16  eighths  ?  24  eighths  ?  32  eighths  ? 
eighths  ?  35  eighths  ?  39  eighths  ?  40  eighths  ? 
eighths  ?  56  eighths  ?  64  eighths  ?  71  eighths  ? 
eighths  ?  100  eighths  ?  800  eighths  ? 

9.  How  many  ones  in  9  ninths  ? 
ninths?      18  ninths?      25  ninths? 
ninths?      36  ninths?      38  ninths? 
ninths?      54  ninths?      58   ninths? 
ninths?     72  ninths?     81   ninths? 
ninths  ?     126  ninths  ? 


1.0  ninths? 

-27  ninths? 

45  ninths  ? 

63  ninths? 
90  ninths? 


33 
46 

80 

17 
30 
50 
69 
100 


10.  How  many  ones  in  10  tenths  ? 


14  tenths?     20 

tenths?  22  tenths?  30  tenths?  36  tenths?  40 
tenths?  45  tenths?  50  tenths?  53  tenths?  60 
tenths?  61  tenths?  70  tenths?  72  tenths?  80 
tenths?  88  tenths?  90  tenths?  99  tenths?  100 
tenths?  150  tenths?  200  tenths  ?  305  tenths?  1000 
tenths  ? 

11.  If  yon  wish  to  divide  17  pounds  of  candy  equally 
among  8  boys,  how  many  pounds  and  what  parts  of  a 
pound  would  you  give  to  each?     If  you  divided  17 
pounds  of  candy  equally  among  9  boys,  what  would 
be  the  share  of  each  (    If  it  was  divided  equally  among 
10  boys,  what  would  be  the  share  of  each  ? 

12.  How  many  eighths  in  5  and  2  eighths  ?     in  20 
and  5  eighths ?    in  3  and  7  eighths  ?    10  and  3  eighths? 
How  many  ninths  in  5  and  2  ninths  ?    10  and  6  ninths  ? 
12   and  8    ninths?     How  many  tenths   in    6    and    3 
tenths  ?    in  25  and  7  tenths?    in  30  and  9  tenths? 

13.  How  much  is  1  eighth  of  3?     Show  by  means 
of  lines  that  1.  eighth  of  3  is  3  eighths  of  1.     Show 
that  1    ninth  of  2  is  2  ninths  of   1.      Show  that  1 
tenth  of  3  is  3  tenths  of  1. 

14.  How  much  is   1  ninth  of  2?    3?    4?    5?    6? 
10?    15?    9?    20?     24?     18?     27?     36?    45?     54? 
63?    72?    75?    81?    88?    90?    100?    900?    1800? 

15.  How  much  is  1  eighth  of  2  ?    3  ?    4  ?    5  ?    10? 
15?    16?    S?     21?     24?     32?     36?     40?    44?    48? 
50?    56?    64  2    72?   80?   96?   97?   100?   800?  1600 


cu  . 

)0? 


FRACTIONS.  85 

1,6.  How  much  is  1  tenth  of  2  ?  3  ?  4  ?  5  ?  6  ? 
10?  12?  14?  17?  19?  20?  30?  35?  48?  60? 
67?  70?  80?  89?  90?  100?  111?  500?  120? 
600  ?  1000  ? 

17.  How  much  is  1 'eighth  of  24  ?     2  eighths  of  24  ? 
3  eighths  of  24  ?     4  eighths  of  16  ?     6  eighths  of  40  ? 
7  eighths  of  32  ?     2  eighths  of  72  ?     10  eighths  of  80  ? 

18.  How  much  is  1  ninth  of  27?     2  ninths  of  27? 

5  ninths  of  27  ?     10  ninths  of  27?     4  ninths  of  36? 

6  ninths  of  72  ?     8  ninths  of  100  ? 

19.  How  much  is  1  tenth  of  50  ?     5  tenths  of  50  ? 
6  tenths  of  50  ?     11  tenths  of  50  ?     3  tenths  of  40?     4 
tenths  of  70?     7  tenths  of  210?     12  tentfis  of  120? 
9  tenths  of  110? 

120.  3  is  one-eighth  of  what  number  ?  One-eighth  of 
hat  number  is  4?  2?  5?  7?  6?  8?  9?  10? 
H  14?  25?  100?  200?  80? 
21.  One-ninth  of  what  number  is  1  ?  2  ?  3  ?  4  ? 
?  6?  7?  8?  10?  12?  19?  22?  31?  100? 
)0?  90? 

22.  One-tenth  of  what  number  is  1  ?    2?    3?    4? 
5?    6?    8?    9?    10?    11?     12?     17?   20?    36?    45? 
100  ?     120  ?     200  ? 

23.  2  is  what  part  of  20?    18?    16?    14?    12?    10? 
8?    6?    4?    2?     5  is  what  part  of  10  ?    5?    20?    15? 
30  ?    25  ?    40  ?    35  ?     45  ?    50  ?     1  is  what  part  of  8  ? 

;10  ?      10  is  what  part  of  80  ?    100  ?    90  ?     100  is 
hat  part  of  100  ?    900?    800?    1000? 

24.  How  do  you  express  in  figures  one-ninth  ?    one- 
snth  ?    one-eighth  ?     five-eighths  ?     six-ninths  ?    four- 
inths  ?  eight  and  three-tenths  ?  nine  and  four-eighths  ? 

o  hundred  and  seven-tenths. 

25.  Eead  the  fractions  |,  -f^-,  f ,  ^  5|,  7-f,  8^-, 
M$. 

26.  The  numbers  above  the  lines  in  a  fraction  are 
ailed  what  ? 

ANS.  The  numerators. 


86  FRACTIONS. 

27.  The  numbers  below  the  line  are  called  what  ? 
ANS.   The  denominators. 

28.  The  denominator  shows  in  how  many  parts  the 
unit  is  divided.     It  denominates  or  names  the  parts. 
And  the  numerator  shows  hovP  many  of  these  parts 
form  the  fraction.     It  numbers  the  parts  taken.     In 
how  many  parts  is  the  unit  divided,  and  how  many  of 
these  parts  are  taken  in  -f  ?    £  ?    f  ?    ^  ?    I  ?    f  •    i ? 

29.  If  we  divide  any  thing  into  11  equal  parts,  what 
is  each  one  of  these  parts  called  ?    Into  12  equal  parts  ? 
Into  20  equal  parts?    Into  35  equal  parts?  Into  60 
equal  parts  ?    Into  100  equal  parts  ? 

30.  How  many  twelfths  in  3  ?    4?    5?    12?     How 
many   twentieths   in    1?    2?    3?    20?      How   many 
Imndredths  in  5  ?    6?    7?    10? 

31.  How  many  ones  in  ff  ^,  fjft,  |f  ? 

32.  How  much  is  -^  of  12  ?    24:?    25  '{    60  ?     How 
much  is  ^5-  of  60  ?    of  90  ?    120  ?     How  much  is 

of  200?    300?    360? 

33.  We  know  now  three  sorts  of  fractions.     1st. 
Proper  fractions,  as  ^,  f ,  f ,  -^,  in  which  the  numera- 
tor is  less  than  the  denominator.     2d.  Improper  frac- 
tions, as  |-,  f,  -2^-,  -{-f ,  -2-lp-,  in  which  the  numerator  is 
equal  to  or  is  greater   than   the   denominator.     3d. 
Mixed  numbers,  as  4f ,  5^-,  7|,  &c.,  made  up  of  whole 
numbers  and  fractions. 

REMARK. — An  improper  fraction  always  stands  for 
either  a  whole  or  mixed  number. 

34.  If  you  increase  the  denominator  of  a  fraction,  do 
you  increase  or  diminish  the  fraction  ?     If  you  increase 
the  numerator  of  a  fraction,  do  you  increase  or  dimin- 
ish the  fraction  ? 


FRACTIONS.  87 


LESSON    IV. 

1.  If  an  orange  costs  9  cents,  what  is  the  cost  of  % 
of  it?     f  of  it? 

2.  If  a  bushel  of  potatoes  costs  60  cents,  what  is 
the  cost  of  ^  of  a  bushel  ?   of  f  of  a  bushel  ?   of  4  of  a 
bushel?   off  of  a  bushel?  of  f  of  a  bushel  ?     What  is 
the  cost  of  TV  of  a  bushel  ?   of  -^  of  a  bushel  ?   of  -J-  of 
a  bushel  ?  of  f  of  a  bushel  ?  of  T^-  of  a  bushel  ?  of  T7T 
of  a  bushel  ?  of  ^  of  a  bushel  ?  of  ¥V  of  a  bushel  ?   of 
fg-  of  a  bushel  ? 

3.  If  an  acre  of  land  costs  72  dollars,  what  is  the 
cost  of  -|  of  the  acre  ?  of  f  of  it  ?  of  ^  of  it?  of  -^ 
of  it?  of  i  of  it?   of  \r  of  it?  off  of  it?  off  of  it?  of 
T*g-ofit?  of  -fa  of  it?  off  of  it? 

4.  If  a  railway  car  travels  40  miles  an  hour,  how 
many  miles  does  it  go  in  ^  of  an  hour  ?  in  -f-  of  an 
hour  ?  in  -|~J  of  an  hour?  in  ^  of  an  hour  ? 

5.  If  a  man  earns  48  dollars  a  month,  how  much 
does  he  earn  in  %  of  a  month  ?  in  f  of  a  month  ?  in  -fa 
of  a  month  ?  in  f  of  a  mouth  ?  in  |-  of  a  month  ?  in 
^  of  a  month?  in  f  of  a  month  ?  in  -^  of  a  month  ? 
in  -/¥  of  a  month  ? 

6.  If  a  man  can  do  a  piece  of  work  in  90  days,  how 
long  will  it  take  him  to  do  £  the  work  ?  ^  of  the  work  ? 
•|  of  the  work  ?  -j3^  of  the  work  ?     How  long  will  it 
take  him  to  do  -fa  of  the  work  ?  f  of  the  work  ?  f  of  the 
work  ?  f  of  the  work  ?   -^  of  it  ?   ^  of  it  ?   f -|  of  it  ? 

7.  84  gallons  of  water  flow  from  a  spring  in  one  hour, 
how  many  gallons  flow  out  in  ^  an  hour '(   in  f  of  an 
hour?  in*f  of  an  hour  ?   in  f  of  an  hour  ?  in  f  of  an 
hour?  in  \%  of  an  hour?  in  ||-  of  an  hour?  in  ^\  of 
an  hour  ?  in  |-f  of  an  hour  ?  in  ^f  of  an  hour  ? 

8.  If  -^  of  an  acre  of  land  costs  5  dollars,  what  is 
the  whole  acre  worth  ?     What  is  -£$•  of  the  acre  worth  ? 
•fa  of  the  acre  ?     5  is  -fa  of  what  number  j 


88  FRACTIONS. 

9.  If  ^  of  a  yard  of  cloth  costs  4  dollars,  what  is  the 
cost  of  the  whole  yard  ?  the  cost  of  |  of  a  yard  ? 

10.  If  20  gallons  of  water  flow  from  a  spring  in  \ 
of  an  hour,  how  many  gallons  flow  in  one  hour  (  in  | 
of  an  hour  ?  in  ±f-  of  an  hour  ? 

11.  If  -J-  of  an  ounce  of  the  otto  of  roses  costs  5 
guineas,  what  is  the  cost  of  1  ounce  ?  of  |  of  an  ounce  ? 
of  -^-  of  an  ounce  ? 

12.  If  one  pound  of  sugar  costs  30  cents,  what  part 
of  a  pound  can  be  bought  for  2  cents  ?  for  3  cents  ? 
5  cents  ?  6  cents  ?  10  cents  ?  15  cents  ? 

13.  If  one  yard  of  ribbon  costs  40  cents,  what  part 
of  a  yard  costs  10  cents  ?  4  cents  ?  8  cents  \  20  cents  ? 

5  cents  ?  1  cent  ?  2  cents  ? 

14.  A  minute  contains  120  half  seconds,  what  part 
of  a  minute  do  12  half  seconds  make  ?  10  half  seconds  ? 

6  half  seconds?  15  half  seconds?  8  half  seconds?  20 
half  seconds?  40  half  seconds?  30  half  seconds?  60 
half  seconds  ?  1  half  second  ?  2  half  seconds  ?  3  half 
seconds  ? 

15.  A  whole  hive  of  bees  yield  36  pounds  of  honey 
in  one  year,  what  part  of  the  hive  yields  6  pounds  ?  9 
pounds  ?  12  pounds  ?  4  pounds  ?  2  pounds  ?  18  pounds  ? 

16.  How  long  will  it  take  a  vessel  which  sails  10 
miles  an  hour,  to  sail  67  miles  ?  45  miles?  102  miles? 
507  miles?  422  miles?  53  miles? 

17.  At  9  dollars  a  barrel,  how  many  barrels  of  flour 
can  be  bought  for  28  dollars  ?  39  dollars  \  69  dollars  ? 
71  dollars  ? 

18.  If  corn  costs  5  dollars  a  barrel,  how  much  can  be 
bought  for  47  dollars?    54  dollars?    61  dollars?  23 


FRACTIONS.  89 


LESSON    V. 

1.  If  5  bushels  of  wheat  cost  9  dollars,  what  is  the 
cost  of  10  bushels  ? 

PROCESS. — 10  bushels  is  2  times  5  bushels,  hence  cysts 
twice  as  much,  or  18  dollars. 

2.  If  12  bushels  of  wheat  cost  20  dollars,   what  is 
the   cost  of  3  bushels  ?     4  bushels  ?     G  bushels  ?     2 
bushels?     1  bushel? 

PROCESS.— 3   bushels  is  ^  of  12  bushels,  and  costs 
erefore  ^  of  20  dollars,  or  5  dollars. 

3.  How  many  dozen  in  10T  ?     112  ?     145  ?     26  ? 

4.  4  men  can  do  a  piece  of  work  in  45  days,-  in  how 
many   days  can    12   men   do    the   same   work  ?     20 
men  ?    40  men  ?     30  men  ?     60  men  ?     32  men  ?     28 

len? 

5..  If  each  yard  of  a  certain  cloth  cost  2^  dollars, 
what  is  the  cost  of  5  yards  ?  Twice  5  and  1  fifth  of  5 
are  how  much  ? 

6.  If  a  man  travels  7  miles  an  hour,  how  far  does 
he   travel   in   1^-   hour?     in  4   and  f  hours?     in  6| 
hours  ?     8  f-  hours  ? 

7.  How  much  is  3  times  10  and  -^  of  10  ?    5  times  10 
and  -&  of  10  ?     8  times  10  and  -&  of  10  ?    10  times  10 
andT8¥oflO? 

8.  How  much  is  9  times  8  and  -|  of  8  ?     10  times  8 
and  |  of  8  ?     7  times  8  and  f  of  8  ? 

9.  At  2  dollars  a  bushel,  what  will  be  the  cost  of 
\\  bushel   of  wheat?     5£  bushels?     100£  bushels? 
25£  bushels  ? 

10.  A  boy  gave  away  7  apples,  which  was  -J  of  the 
number  of  apples  which  he  had ;  how  many  apples 
had  he  ? 

11.  Frank  is  10  years  old  and  is  -J-  as  old  as  his  father, 
how  old  is  his  father  ? 

12.  If  |  of  an  acre  of  land  costs  25  dollars,  what  is 


90  FRACTIONS. 

the  cost  of  -|-  of  the  acre  ?     Of  the  whole  acre  ?     Of  -f- 
of  the  acre  ? 

13.  If  T9¥  of  a  gallon  of  water  leaks  from  a  vessel  in 
27  hours,  in  how  many  hours  will  -fa  of  a  gallon  leak 
out.?     In  how  many  hours  will  one  gallon  leak  out  ? 
-fa  of  a  gallon  ?     -^f  of  a  gallon  ? 

14.  Two  boats  are  moving  in  the  same  direction  ; 
one  gains  on  the  other  %  of  a  mile  in  35  minutes  ;  in 
how  many  minutes  does  it  gain  -J-  of  a  mile  ?     1  mile  ? 
f  of  a  mile  ? 

15.  A  merchant  gains  ^  of  a  dollar  in  each  sale  of 
8  dollars  ;  what  sales  must  he  make  in  order  to  gain  -J- 
of  a  dollar  ?     1  dollar  ?     £  of  a  dollar  \ 

16.  8  is  |^  of  what  number?     Is  -J  of  what  number? 
-ig9-  of  what  number  ?     ^  of  what  number  ?     f  of  what 
number  ?     |-  of  what  number? 

PROCESS. — If  8  is  -£  of  the  number,  then  ^  of  8,  or 
2,  is  ^  of  the  number.  If  \  of  the  number  is  %,%of  it, 
or  the  entire  number,  is  6  times  2,  or  12. 

17.  3  is  f  of  what  number?     Is  f  of  what  number? 
if-  of  what  number  ? 

18.  8  is  f  of  what  number?     £  of  what  number? 
-^  of  what  number?     T\-  of  what  number?     yf^j-  of 
what  number  ?    -^  of  what  number  ?    -|  of  what  num- 
ber ? 

19.  12  is  f  of  what  number?     -|  of  what  number? 
-ff  of  what  number?     |f  of  what  number?     -j^-  of 
what  number  ? 

20.  9  is  f  of  what  number?     |-  of  what  number? 
-^  of  what  number  ?     T9g-  of  what  number  ?     -|  of  what 
number  ?      ^  of  what  number  ? 


FRACTIONS. 


91 


LESSON    VI. 


lV       IjJT       A       "A       T6¥       TV       TIT       A       A 

1.  If  a  line  be  divided  into  two  equal  parts,  and 
of  these  parts  be  divided  into  two  equal  parts. 


92  FRACTIONS. 

into  how  many  equal  parts  will  the  line  be  divided  ? 
One  half  is  equal  to  how  many  fourths  ? 

2.  If  a  line  be  divided  into  two  equal  parts,  and 
each  of  these  parts  into  three  equal  parts,  what  part  of 
the  whole  line  will  each  of  those  parts  be  ?     One  half 
is  equal  to  how  many  sixths  ? 

3.  If  we  divide  each  half  of  the  line  into  4  equal 
parts,  how  many  of  these  parts  will  the  whole  line  con- 
tain ?     %  is  equal  to  how  many  eighths  ?     ^  is  equal  to 
how  many  tenths  ? 

4.  In  what  fractions  can  halves  be  expressed  ? 
ANS.  In  fourths,  sixths,  eighths,  tenths,  cfe.,  whose 

denominators  are  multiples  of  2. 

5.  If  a  line  be  divided  into  three  equal  parts,  and 
each  of  these  parts  successively  into  2,  3,  4,  5,  &c., 
equal  parts,  what  part  of  the  line  will  each  of  these 
parts  be  respectively  ? 

6.  How  many   sixths,   ninths,  twelfths,   fifteenths, 
twentieths  ? 

7.  In  what  equivalent  fractions  can  thirds  be  ex- 
pressed ? 

ANS.  In  sixths,  ninths,  twelfths,  &c.,  whose  denomi- 
nators are  multiples  of  3. 

8.  If  a  line  be  divided  into  4  equal    parts,  and 
each  of  these  parts  into  2  equal  parts,  in  how  many 
parts  will  the  line   be  divided  ?     %  is  equal  to  how 
many  eighths  ?     How  many  sixteenths  ?     How  many 
twelfths  ? 

9.  One   fifth   is  how    many   tenths  ?     How   many 
twentieths  ?     How  many  fifteenths. 

10.  One  sixth  is  how  many  twelfths  ?  How  many 
eighteenths  ?     How  many  twenty-fourths  ? 

11.  What  are  the  denominators  of  equivalent  frac- 
tions in  which  fourths  may  be  expressed  ? 

12.  What  are  the  denominators  of  equivalent  frac- 
tions in  which  fifths  may  be  expressed  ? 

13.  What  are  the  denominators  of  equivalent  frac- 
tions in  which  sixths  may  be  expressed  ? 


FRACTIONS.  93 

14.  Express  £  in  parts  of  a  unit  3  times  as  small. 
In   parts  5  times  as  small.      10  times  as  small.     In 
parts  100  times  as  small. 

15.  Express  f  in  parts  5  times  as  small.     Find  four 
fractions,  each  of  which  is  equal  to  f .     f  is  equal  to 
how  many  forty-fourths? 

16.  How  many  parts  of  a  unit,  10  times  as  small,  is 
%  equal  to  ?     How  many  parts  of  a  unit  10  times  as 
small  is  ^  equal  to  ?     Find  5  fractions,  each  equal  to 
•f .     f  is  equal  to  how  many  sixtieths  ? 

17.  Find  10  fractions,  each  of  which  is  equal  to  f. 
How  many  parts  of  a  unit  15  times  as  small  is  -J-  equal 
to  ?     Is  f  "equal  to  ?     Find  6  fractions,  each  of  which 
is  equal  to  -| . 

18.  1  =  •£  =  ^  =  ^  =  \  =  \=  \  ===T^  =  -gV  = 

1  _    2    __   _     £     JLQ.Q. 

•4   Q  n       -    —       ft  o  0'    —  T     ~  !      • 

(Read  1  =  how  many  halves,    &c.,    &c.  ?      1=2 
what?     3  what?  &c.) 

1  Q       Q !   « ?    _          !  f          _   12.  16. JJ fifi. 

1J.     ^ -g- 3- ¥ -j-g- TO"0~~  I  ?' 

90     q  ?  «    _          t  -I    & 12 4i 

^U.    O         =    -3  T7 T¥Q- -3-^- ,  - 

21.  What  multiples  must  the  denominators  of  frac- 
tions be  in  which  sevenths  can  be  expressed  ?      In 
which  eighths  can  be  expressed  ?     In  which  ninths  can 
be  expressed  ?     In  which  tenths  can  be  expressed  ? 

22.  Give  5  fractions,  equal  to  \.      Give  5  equal 
to   |.      Give  5  fractions  equal  to  •£.      Give   5  equal 
to  TV 

23.  Express  f  in  parts  4  times  as  small.     Express 
f  in  parts  6  times  as  small.     T9F  is  equal  to  how  many 
twentieths  ?    how   many   fiftieths  ?    how   many  hun- 
dredths  ?  how  many  two  hundredths  ?     |  is  how  many 
forty-eighths  ? 

24.  To  change  fifths  into  tenths,  into  how  many 
equal  parts  must  each  fifth  be  divided  ?     To  change 
fifths  into  fifteenths,  into  how  many  ?  into  twentieths, 
into  how  many  ?    into  thirtieths,  into  how  many  must 
each     be  divided  ? 


94  FRACTIONS. 

25.  T  f-  are  how   many  forty-eighths  ?     how   many 
sixty-fourths  ?    how  many  ninety-sixths  ? 

26.  Change  f,  y^,  y7-^,  and  -J-J-,  each  to  thirtieths. 

27.  Change  f,  f,  If,  and  ^  each  to  ninetieths. 

28.  Change  f  ,  f  ,  f  ,  and  f  to  tenths. 

29.  Change   y^   to   three-hundredths.     Change  T5¥ 
to  thirty-ninths.     Change  -J  to  ninety-sixths. 

30.  How  many  72ds  in  f  ?     f?     f  ?     f?     1?     -U-? 

fl?     H»     *** 

31.  How  many  IGOths  in  £  ?    f  ?    £  ?   f  ? 


32.  How  many  lOOths  of  a  dollar  make  y1^  of  a  dol- 
lar ?     How  many  half  a  dollar  ? 

33.  How  do  you  find  a  fraction  of  a  greater  denom- 
inator equal  to  a  given  fraction  ? 


LESSON    VII. 

1.  A  line  is  divided  into  10  equal  parts;  what  part 
of  the  line  will  T5^  make  ?     If  a  line  is  divided  into  12 
equal  parts,  what  part  will  T6^-  make  ? 

2.  Reduce  A;    *5    A;    i£ ;    WV;    W;  to  halves. 
What  must  be  done  in  order  to  bring  T6^-  back  to  -J-  ? 

3.  Change  ^  to  fifths.    Change  |-f-  to  fifths.    Change 
-^  to  halves.     Change  |-|-  to  thirds ;  also  to  fourths. 

4.  T82-  of  a  bushel  is  how  many  sixths  of  a  bushel  ? 
How  many  thirds  of  a  bushel  ? 

5.  Express  ^-  in  parts  of  a  unit  twice   as   great. 
Express  it  in  parts  of  a  unit  5  times  as  great.     Ex- 
press it  in  parts  of  a  unit  10  times  as  great. 

6.  |-f  is  equal  to  what  fraction  with  a  denominator 
half  as"  great  ?     One-third  as  great  ?     £  as  great  ?     -J- 
as  great  2     -|-  as  great  ?     T^-  as  great  ?     -g^  as  great  ( 

7.  How  do  you   change  a  fraction  into  an   equal 
fraction  of  smaller  denominator  ?     In  order  to  express 
a  fraction  in  parts  3  times  as  great,  the  numerator  and 


FRACTIONS.  95 

denominator  must  be  divisible  by  what  ?  In  parts  4 
times  as  great,  divisible  by  what  ?  In  parts  6  times 
as  great  ? 

8.  Can  -f-,  f,  f 3  f,  |,  TV  be  expressed  exactly  in 
greater  parts  of  a  unit  ?     Can  they  be  expressed  in 
smaller  parts  of  a  unit  ? 

9.  ^  =  f  =  |-  =  1-g-g-,  &c.     |-  is  said  to  be  in  lower 
terms  than  any  of  its  equivalent  fractions,  being  ex- 
pressed in  smaller  figures.     In  the  same  manner  -f-  is 
a  fraction  at  its  lowest  terms,  or  simplest  form. 

10.  When  is  a  fraction  in  its  simplest  form  ? 

ANS. —  When  its  numerator  and  denominator  have 
no  common  divisor  greater  than  1. 

11.  Reduce  f  f  to  its  simplest  form. 

12.  Reduce  -J-f  to  its  simplest  form. 

13.  Reduce  f|-  and  y2^  each  to  its  simplest  form. 

14.  Reduce  -f|-  to  its  simplest  form. 

15.  Reduce  |f  to  its  simplest  form. 

16.  Reduce  -f/-^  to  its  simplest  form. 

17.  How  do  you  reduce  a  fraction  to  its  simplest 
form  ? 

18.  If  you  multiply  the  denominator  of  the  fraction 
f  by  4,  what  change  do  you  make  in  the  value  of  the 
fraction  \     If  you  multiply  the  numerator  of  the  frac- 
tion f  by  4,  what  change  do  you  make  in  the  value? 

19.  If  you  multiply   both  the  numerator  and   de- 
nominator of  f  by  4,  the  fraction  | •  *  |  =  =  | ;  why  ? 

ANS. — Because  we  take  4  times  as  many  parts •,  each 
of  which  is  one  fourth  as  great. 

20.  If  you  divide  the  denominator  of  the  fraction  T8j 
by  4,  what  change  of  value  do  you  make  in  the  frac- 
tion ?     If  you  divide  the  numerator  by  4  what  change 
do  you  make  ?     If  you  divide  both  numerator  and  de- 
nominator by  4  you  make  no  change  ;  why  ? 

ANS. — Because  you  take  one  fourth  as  many  parts, 
each  of  which  is  4  times  as  great. 

NOTE. — The  teacluT  should  ilhr^trato  this  with  other  examples  until 
it  is  perfectly  clear  to  each  punil. 


96  FRACTIONS. 

21.  How  many   9ths  in  -||  ?     How  many   7ths  in 

22.  How  many  8ths  in  f  £  ?     In  f  f  ? 

23.  Eeduce  to  the  lowest  terms  the  following  frac- 
3ns : 


tions : 

A? 

*-  i.         VJ»  1  T  V-> 

lower  terms, 


24.  Give  tive  fractions  which  cannot  be  reduced  to 
>wer  terms. 


LESSON    VIII. 


1.  How   do   you   express  %  and   -J-   by   two   other 
fractions  having  tKe  same  denominator  —  how  do  you 
proceed  ? 

ANS.—  £  —  f  —$  =  |  =  TV  <fic.  t  =  f  =  |  —  A, 
dec.,  a*#  ^<3  6'^;?  express  either  ^  oy  f  <m6?  -J  5?/  f  ?  ^^  i 
^y  yV?  a?7y^  i  fy/  1^5  and  g°  °n  higher  and  find  others. 

2.  What  is  the  6  or  12  called  in  this  case  ? 

ANS.  —  77/^  common  denominator  ',  a;i<^  mw6'^  J^  # 
common  multiple  of  2  an^  3. 

3.  What  is  the  least  common  denominator  which 
any  two  fractions  equal  to  ^  and  ^  can  have  ? 

ANS.  —  6,  £/^  least  common  multiple  of  2  and  3. 

4.  What  is  the  least  common  denominator  of  ^  and 
i?     Of  I-  and  |?     Oft  and  i?     Of^andf? 

5.  What  are  6  of  the  common   denominators  of  f 
andf? 

PKOCESS.  —  6  common  multiples  of  3  and  4  are  12, 
24,  36,  48,  60,  72.  Hence  these  are  6  of  the  common 
denominators  of  fractions  equal  to  f  and  f  respectively. 

6.  What   is   the  least  common  denominator   of  f 
and  f  ? 

7.  Reduce  the  fractions  f  and  £  to  the  least  common 
denominator. 

PROCESS.  —  The  least  common  multiple  of  8  and  5  is 

JA     3   —    15 

^u         — 


FRACTIONS.  97 

8.  Reduce  ^,  ^,  and  £  to  the  least  common  denomi- 
nator. 

9.  What  is  the  least  common  denominator  of  f  and 

A? 

10.  What  is  the  least  common  denominator  of  -§-  and 

A* 

11.  What  is  the  least  common  denominator  of  -^  and 

_9_2  * 

10* 

12.  What  is  the  least  common  denominator  of  T4T 
and  ^  ? 

13.  What  is  the  least  common  denominator  of  f  and 

A? 

14.  What  is  the  least  common  denominator  of  ^,  J-, 
and  f  ? 

15.  What  is  the  least  common  denominator  of  ^ 

ru>  and  i  2 

16.  What  is  the  least  common  denominator  of  -|-,  T^-, 
and  -^  ? 

Reduce  to  common  denominators  — 

17.  f  and  f.  20.  TV  and  -^. 

18.  f  andf  21.  1  and  ^ 

19.  f  and  f.  22.  ||-  and  ^. 

23.  Reduce  to  the   least   common   denominator  % 
and  TV 

24.  Reduce  to  the  least  common  denominator  y1^ 
and  ^-. 

25.  Reduce  to  the  least  common  denominator  £?  T3¥, 
and  -J-. 

26.  Reduce  to  the  least  common  denominator    ,    5 


27.  Reduce  ^,  -j^,  and  ^  to  the  least  common  de- 
nominator. 

28.  Reduce  £,  f  ,  f,  |,  -j^-  and  ||-  to  the  least  coin- 

«>n  denominator. 
29.  How  do  you  reduce  fractions  to  common  de- 
minators  ? 

30.  How  do  you  reduce  fractions  to  the  least  com- 
7 


98  FRACTIONS. 

mon  denominators,  when  the  denominators  have  no 
common  factor? 

31.  How  do  yon  find  the  least  common  denominator, 
when,  the  denominators  have  a  common  factor  ? 

LESSON    IX. 

1.  A  boy  gave  his  brother  ^  of  an  apple,  and  his 
sister  £  of  an  apple  ;  how  many  fourths  did  he  give  to 
both? 

2.  A  man  sold  a  bushel  of  apples  for  -|  of  a  dollar, 
a  bushel  of  potatoes  for  |  of  a  dollar,  and  a  bushel  of 
meal  for  -|  of  a  dollar.     How  many  dollars  did  he  re- 
ceive for  all  ? 

3.  A  farmer  divided  his  land  into  several  lots.     One 
lot  contained  |  of  an  acre,  a  second  lot  ^  of  an  acre, 
a  third  lot  -f-  of  an  acre,  and  a  fourth  lot  if-  of  an  acre. 
How  many  acres  in  all  the  lots  ? 

4.  One  man  pays  me  8|  dollars,  and  another  pays 
me  1-J-  dollar  ;  how  much  do  both  together  pay  me  ? 

5.  |   and  f   are  how  much?      $  +  f  ?      J/  +  *£•  ? 

T9o 


7.  A  workman  receives  from  his  employer  at  dif- 
ferent times  if  of  a  dollar,  4^-  of  a  dollar,  3^  of  a 
dollar,  11T5T  of*  a  dollar,  and  2y9^-  of  a  dollar.     How 
many  dollars  did  he  receive  altogether  ? 

8.  A  vintner  has  in  his  cellar  4  casks  ;  the  first  con- 
tains Sy1^-  gallons  of  wine,  the  second  7T6Q-,  the  third 
•f-^,  and  the  fourth  5^  gallons  ;  how  many  gallons  of 
wine  in  the  four  casks  \ 

9.  f-f1L  =  ?     Can  these  be  added  without  changing 
them  first  into  the  same  denomination  ? 

PROCESS.—  J—  f  ;  £  —  |  ;  £  +  |=f.+_f  =  |. 

10.  How  do  you  add  fractions  having  a  common 


FRACTIONS.  99 

demominator?     Plow  do  you  add  fractions  of  different 
denominators? 

11.  i+|-»  f+t 


13.  . 

PROCESS.  —  1|-  i'<s  f.      Hence  we   add  J-  a^<#  -J-,  or 


.  4i-+f=?     3f+f=?     10  +  6^4-5-1-=? 


15.  What  is  the  sum  of  14|,  10£,  and  5J  ? 

16.  What  is  the  sum  of  6^,  8|,  and  5f  ? 

17.  What  is  the  sum  of  ^  12|,  15},  and  9|  ? 

18.  Add  f  ,  3J,  4,  and  5f. 

19.  A  man  planted  2^-  acres  in  corn,  sowed  lOf- 
acres  in  oats,  and  5f  acres   in  wheat.     How  many 
acres  had  he  in  corn,  wheat,  arid  oats  ? 

20.  I  bought  2|-  bushels  of  potatoes  of  one  man,  6f 
bushels  of  another,  and  4}  bushels  of  a  third.     How 
many  bushels  did  I  buy  from  the  three  men  ? 

LESSON   X. 

1.  I  purchased  a  lot  containing  |  of  an  acre,  and  cut 
off  from  it  a  lot  containing  |  of  an  acre.     How  many 
fourths  of  an  acre  did  the  other  part  contain  ? 

2.  Mary  had-|  of  an  orange,  and  gave  f  away. 
How  much  had  she  left  ? 

3.  Charles  had  a  melon,  and  gave  T3¥  of  it  away. 
How  much  had  he  left  ? 

4.  Julian  had  5  oranges,  and  gave  away  f  of  one  of 
them.     How  many  had  he  left  ? 

5.  From  •£$  take  -f$.     From  y2^  take  yW.     From 
take  TV     From  18f  take  13|,     From  lOf  take  6f  . 

6.  How  much  is  1  less  $  ?     2  less  -J-  ?     3  less  f  ?     15 
lessf?    16  less  2-J-  ?    5  less  If?    10  less  TV?    5  less  4  f? 


H 


100  FRACTIONS. 

T^S 

7.  How  much  is  £  less  ^  ? 
SOLUTION.— |- 1==|-  f  —  f 

How  much  is  |  less  ±  ?     -f  less  f  ?     -f-  less  |  ?     f  less 
1?     flessj-? 

8.  What  is  the  difference  between  ^  and  |  ?     -|  and 
A»     «  a*d  ft?     A  and -ft »     f  and -^  ? 

9.  From  3£  take  2f . 

PROCESS.-^-  2f  _  y-  *  _  ft-  ff  -.  A- 
From  3|  take  2£.     How  much  is  3|— 1|  J     What 
is  the  difference  between  lOf  and  9|-  ? 
How  much  is — 

10.  3J-2J?  13.  7J~  4f? 

11.  4|-2f?  14.  9-i—  8f  ? 

12.  3f--lf  ?B  15.  12-i|—  5f 

16.  What  is  the  difference  between  19  and  ^Q?   Be- 
tween 12  and  f£  ?     Between  16  and  1|? 

17.  ^  +  ^  is  how  much  less  than  1  ? 

18.  -|-  +  f  +  f  is  how  much  less  than  2? 

19.  i  sold  ^  of  my  land  to  one  man,  ^  of  it  to  an- 
other, and  -^  of  it  to  a  third.     How  much  of  it  did  I 
have  left  ? 

20.  I  conceive  two  numbers,  one  is  J-  and  the  other 
is  f  greater — what  is  the  number?     What  number  is 
f  less  than  f  ? 

21.  Of  two  numbers  f  is  the  greater  and  -J-  is  the 
difference  between  them — what  is  the  smaller  ? 

22.  How  much  greater  is  %  than  f  ? 

23.  How  do  you   subtract   one   fraction  from  an- 
other ? 

24.  How  do  you  subtract  a  fraction  from  a  whole 
number  ? 

25.  How  do  you  subtract  one  mixed  number  from 
another  ? 


FRACTIONS.  101 


LESSON    XI. 

1.  "What  is  the  cost  of  5  pounds  of  tea  at  £  of  a 
dollar  a  pound  ?    5  times  three-fourths  is  how  many 
fourths?     |+i+|.+|.+i4i| 

2.  If  |-  of  a  barrel  of  flour  last  a  family  one  month, 
how  much  flour  will  last  the  family  6  months?     f  +  f 

+*+*+*+*—?     6  times  f  —  ? 

3.  How  much  is  15  times  -|?    5  times  1?    11  times 

I? 

4.  How  much  is  5  times  f  ?    19  times  f  ?    11  times 
|  ?    26  times  f  ?    14  times  f  ?    25  times  f  ?    10  times 
|?    15  times  |?    20  times  f?^-?    £?  •&?    A?     24 
times  |?    f?    I?    I?    |?    1?>TV? 

5.  What  is  meant  by  multiplying  5  by  ^-? 
ANS. — To  take  ^  of  5. 

6.  If  a  number  be  multiplied  by  a  number  less  than 
unity,  is  the  product  greater  or  less  than  the  first 
number  ?     Multiply  5  by  f-. 

7.  How  much  is  10  times  f  ?     f  xlO?     f  of  10? 
How  much  is  32  times  f?    |x32?    f  of  32?     How 
much  is  23  times  f  ?    f  x  23  ?    f  of  23  ? 

8.  How  do  you  find  a  given  fraction  of  any  number  ? 

9.  A  man  sold  10  yards  of  cloth  at  2-J-  dollars  a 
yard ;  what  did  he  receive  for  the  10  yards  ? 

10.  At  2  dollars  a  yard,  what  will  3-J-  yards  of  cloth 
cost? 

11.  At  5  dollars  a  barrel,  what  will  4£  barrels  of 
flour  cost? 

12.  If  a  man  walks  5  miles  an  hour,  how  far  will 
he  walk  in  6|-  hours  ?       «> 

13.  How  much  is  5  x  6|?    6  x  7-J?    6f  x  6?    9f 
X62     4^x3?    1^x4?     5x4A»     31^x10? 

x  9?    91|x  8?    20f  x  Y? 

14.  How  much  is  12  times  10  and  T%  of  10? 

15.  How  much  is  9  times  7  and     of  7? 


102  FRACTIONS. 

16.    10XTV=  ?      ^Xl0=?       -^oflO  —  ?      11  X 


•?     Wxl2? 

17.  How   much   do   you  get   when   you    multiply 
i  by  1  ? 

18.  How  much  do  you  get  when  you  multiply  -J- 
by  ^  ?     How  much  is  -J-  x  \  ? 

ANS.— %  of  \. 

19.  If  you  take  a  line  and  divide  it  into  3  equal 
parts,  and  take  ^  of  one  of  these  parts,  what  part  of 
the  whole  line  do  you  take  ?     If  you  divide  the  line 
into   two  equal   parts,  and   take  -J-  of  one  of  these 
parts,  what   part   of  the   whole   line  do  you    take? 
i  x  |-  =  %  of  -J-  =  -J-  of  |-.     Show  this  by  a  line  on 
your  slate. 

°*   TT  ===  T7T  X  y-Q-  =  . 

21.  f  multiplied  by  f  is  2  times  -J-  x  J  =  ?     How 
much  is  I-  off?    Off?    Of-jV?    Of^?    Of||? 

PROCESS. — -J  x  f,  o/1  -^  of  ^  is  3  x  ^  ^/"^-  =  ^C-. 

22.  How  much  is  I- of -§-?     |of|?     fof|? 
How  much  is  |  of  ^  ?     -J-off?     j-off?     |  of  *f  ? 
of  |? 

23.  How  much  is  |  xf  ?     fx  |  =  f  of  f  —  fof  f  ? 
How  much  is  -f  x  f  ?     -f-  x  -^  ?     yV  x  iV  -     I  x  I  • 

4X3?       |X|?       |X3?       ^x,f|       f><7? 

24.  How  much  is  -f-  x  |  ?     ^  of  f  ?     f  of  |  ?     ^ 


25.  A  man  sold  a  -bushel  of  wheat  for  1£  dollars. 
What  is  the  cost  of  -J-  of  the  bushel  ? 

26.  How  much  is  1£  multiplied  by  J  ?     What  does 
it  mean  to  multiply  1|-  by  -J  ? 

ANS.  —  To  take  -i-  of  ^^.     PROCESS.  —  1^  is  f  .     %  of 


27.  Sold  a  barrel  of  flour  for  7|  dollars.     What  was 
the  cost  of  -§-  of  the  barrel  ? 


FRACTIONS.  103 

28.  How  much  is  8$  x  |  =  |  of  8£  ?     How  much 
is  2±  x  i?     9i  x  f?     3£  x  | ?     4f  x  f  ?     If  x  3TV? 
fx3f?     JxlOi? 

29.  What  is  the  cost  of  3£  yards  of  cloth  at  1%  dol- 
lar a  yard?     How  much  is  3£  x  1J? 

SOLUTION.— 3£  x  1£  =\3-  x  f  =  f  of  *£-  — ^«-  or  4f 

30.  How  much  is  If  x  1£?     1J  x  2f  ?     2f  x  7J-? 
2fxl£?     6fx2TV?     Hx6f?     2|x3£?1 

31.  Find  the  sum,  difference,  and  product  of  f  and  -J. 
Of  1|  and  2£.     Of  2J-  and  3f .     Of  1£  and  2f     Of  7| 
and  3f 

32.  How  do  you  multiply  a  fraction  by  a  whole 
number  ?     How  do  you  multiply  a  fraction  by  a  frac- 
tion, or  take  a  part  or  parts  of  a  fraction  ?     How  do 
you  multiply  a  mixed  number  by  a  mixed  number  ? 

33.  A  man  bought  a  ton  of  coal  for  17f  dollars  a 
ton.     What  was  the  cost  of  -J-  of  a  ton  ? 

34  A  man  bought  10  bushels  of  wheat  for  27|  dol- 
lars. What  did  he  pay  for  each  bushel  ? 

35.  5  yards  of  cloth  cost  18f  dollars.  What  is  the 
cost  of  one  yard  2 

LESSON   XII. 

1.  If  an  apple  is  divided  into  two  equal  parts,  each 
part  is  ^.     1  divided  by  2  =  ?     If  2  apples  each  be 
divided  into  3   equal  parts  and  ^   part  of  each  be 
taken,  we  will  .have  f  of  an  apple.     2  divided  by  3 
—  what  ? 

2.  What  do  we  mean  by  dividing  1  by  2  ?    2  by  3  ? 
ANS. — To  divide  1  into  two  equal  parts  and  take 

one  of  them,  which  gives  ^.     In  the  same  manner  to 
divide  2  ly  3  gives  f . 

3.  What  fraction  is  equal  to  1  divided  by  5  ?    l-~6  ? 
7?     i^s?     1-^-9? 

4.  If  you  divide  half  an  orange  into  two  equal 
parts,  what  part  of  the  orange  do  you  get  ?     £  divided 
by  2  is  what  ?     .  • 


104  FRACTIONS. 

5.  How  much  is  £  divided  by  3  ?    4  ?     5  ?     6  ?     7  ? 
8  ?     9  ?     10  ? 

ANS. — J-  divided  l>y  3  is  ^  of  -J-,  or  -|-,  cfee. 

6.  How  much  is  £  divided  by  3?    4?    5?     How 
much  is  f  divided  by  3?     4  ?     5  ?     6?     7?     8? 

7.  A  man  had  |-  of  a  barrel  of  flour,  and  divided  it 
equally  among  6  persons.     How  much  did  he  give  to 
each  ?     How  much  is  ^^6  =  -J-  of  £  ? 

8.  A  person  wished  to  divide  f  of  an  acre  of  land 
into  10  equal  lots  ?     How  much  land  would  each  lot 
contain  ? 

9.  Divide  £  by  9.  By  2.  By  5.  By  7.  By  6.    Divide 
|  by  11.     £J  by  5.     fl  by  4.     ff  by  5. 

10.  How  many  halves  make  1  ?     How  often  is  % 
contained  in  one  ?     How  many  halves  in  2  ?     How 
often  is  £  contained  in  2  ? 

11.  A  person  wishes  to  distribute  4  barrels  of  flour 
so  as  to  give  ^  of  a  barrel  to  each  of  a  number  of 
families.     How  many  families  could  he  supply  ? 

12.  How  many  thirds  in  10  ?     -J-  is  contained  in  10 
how  many  times  ?     £  is  contained  in  5,  6,  7,  8,  9,  how 
many  times  ? 

13.  How  much  is  1  divided  by  -J-  ?     2  divided  by 
-J-  ?     5  divided  by  -J-  ?     9  divided  by  |  ?     11  divided  by 
-iV  ?     12  divided  by  ^  ? 

14  How  much  is  10  divided  by  ^  ?  f  is  contained 
in  10  how  many  times  ? 

SOLUTION. — J-  is  contained  in  10,  50  fo'm<?«s  /  hence 
f  is  contained  half  as  many  times,  or  25  fa'ratfs. 

15.  How  much  is  5  divided  by  -|  ? 

SOLUTION. — 5  divided  l>y  \  gives  45  ;  hence  5  di- 
vided by  -|  must  give  a  quotient  6  times  as  small 

=  4/  =  7f 

16.  How  much  is  6  divided  by  f  ?     6  is  f  of  what 
number  ?    How  much  is  6  divided  by  5  ?    6  is  5  times 
what  number  ?     How  much  is  10  divided  by  -f-  ?     10 
is  %  x  by  what  number  ?     10  is  -f  of  what  number  ? 


FRACTIONS.  105 

17.  How  much  is  6-f-f  ?     6  is  f  of  what  number  ? 

18.  How  much  is  4-^-J-  ?     4  is  ^  of  what  number  ? 

19.  How  much  is  16-^f  ?  16  is  |  of  what  number  ? 

20.  How  much  is  12-^  ?     10-Hr  ?     6  divided  by 
|  ?     9-T--&  ?     5  divided  by  f  ?     11  divided  by  ^  ? 

21.  How  much  is  100  divided  by  f  ?     |  ?     f  ?     £  ? 

22.  2  is  f  of  what  number  ?    5  ?    3  ?    7  ?     10  ?     11  ? 
12?     15? 

23.  60  is  f  of  what  number  ?     f  of  what  number  ? 
-f  of  what  number  ?     T^-  of  what  number  ? 

24.  How  do  you  divide  a  fraction  by  a  whole  num- 
ber ?     How  do  you  divide  a  whole  number  by  a  frac- 
tion? 

25.  If  one  soldier's  ration  of  flour  is  1-J-  pound,  how 
many  rations  will  8  pounds  of  flour  make  ?     1^-  is  con- 
tained how  often  in  8  ? 

SOLUTION. — 1-J-  is  |-,  and  8-f-f  is  6. 

26.  How  often  is  6f  contained  in  9  ?     How  much  is 
60  divided  by  3|-?     By  1-JJ     3f  ?     How  much  is  2 
divided  by  3^  ?  4^  ?     How  much  is  5  divided  by  4£  ? 

^27.     How  much  is  2f  +•  2  ?    8f  -f-  7  ?    14f  -f-  12  ? 

28.  If  f  of  a  ton  of  coal  is  burned  in  a  stove  in  one 
month,  how  long  will  f  of  a  ton  last  for  the  same 
stove?     f  is  contained  how  often  in  f  ?     3  in  9  how 
many  times  ?     -jf-  in  •§•  how  many  times  ?     -f$  in  T9¥  ? 
-A-  in  69  ? 

29.  f  in  -25°-  how  many  times  ?     -^  divided  by  f  is 
how  much?     How  much  is  -i/-  divided  by  f  ?     -11^ 
contains  ^  how  many  times?     -i|-Q--f-^-=  ?     -^^-v- 
14  =  ?     -^gQ-~|-g=  ?     J^--^.|=?      «.  contains  -|  how 
many  times  ?     f  divided  by  f  is  how  much  ?     6  con- 
tains 5  how  often  ?     5  divided  by  6  is  how  much  ? 

30.  1  divided  by  ^  is  how  much  ?     |-  divided  by  -^ 
is  how  much  ? 

ANS. — Since  1  -=-  ^  is  3,  |-  -f-  -J  ?m^  l)e  ^  of  3,  0r  f . 


106  FRACTIONS. 

31.  How  much  is  £  divided  by  -J-  ? 

Asrs.  —  J-  =  ^  <md  -J-  =  ^.;^--f-^5.=36  divided  ly 

5  —  I  =  H- 

32.  How  much  is  J-  divided  by  \  ? 

PROCESS.  —  £-^-J-=  |.    Hence,  ^~\  _  5  ^'m0s  f  =  -1/- 

33.  How  much  is  %  divided  by  |-  ? 

PROCESS.  —  |  divided  by  ^  is  f  ,  <mrf  %  divided  ~by  f 
^tf  be  |  </  f  —  ^. 

34.  How  much  is  f  divided  by  f  ? 

ANS.  —  f  divided  by  ^  is  ^  and  f  divided  by  |-  ^  £ 


. 

35.  How  much  is£-^-==f  x|? 

36.  How  much  is  -f^-f^  =&  x  ^  ? 

37.  How  much  is  -^-J  =  T\  of  f  =T33-  x  f  ? 

38.  How  much  is  £  divided  by  f  ?     by  f  ?     f 

f»     A? 

39.  How  much  is  f  divided  by  TV  ?    by  f  ?    |  ? 

40.  How  much  is  |  divided  by  TV  ?     |  ?     f  'i     f  ? 

41.  How  do  you  divide  one  fraction  by  another  ? 

42.  A  tailor     having  8f  yards  of  cloth,  cut  it  into 
pieces,  each  2|-  yards   long  —  how  many  such  pieces 
did  he  get  ?     2J  is  contained  how  often  in  8|  ? 

PROCESS.  —  2£  is  f,  8f  is  -^  -^  divided  ~by  f  =-a/-  x 


.- 

43.  How  much  is  1|-  divided  by  2f|    24-  divided  by 
1|-  ?    If  divided  by  If  ?    If  -f-  2f  ?    6|  Vlf 

44.  How  much  is  2^  divided  by  64  ?    5|-  divided  by 
4f?    5|-  divided  by  2f  ?    •£  divided  by  3|  ?    4f  divided 


45.  If  one  bushel  of  wneat  cost  f  of  a  dollar,  what 
is  the  cost  of  -f-  of  a  bushel  ?     -i^-  of  a  bushel  ?     5£ 
bushels  ?    lOf  bushels  ?    If  bushel  ? 

46.  If  one  bushel  of  wheat  cost  f  of  a  dollar,  how 
much  wheat  can  you  buy  for  -|  of  a  dollar?   -^  of  a 
dollar?    5£  dollars?    10-}  dollars?    If  dollar? 

NOTE.  —  Let  the  pupil  make  5  problems  like  the  above,  and  solve 
them. 


FRACTIONS.  107 

47.  Add  %  and  -J,  and  multiply  the  sum  by  f . 

48.  From  3£  take  f,  and  divide  the  remainder  by  f . 

49.  Multiply  £  by  1^,  and  divide  the  product  by  f. 

50.  Multiply  2^  by  J-,  and  divide  the  product  by  4^. 

51.  Is  the  dividend  always  larger  than  the  divisor? 
Is  the  quotient  always  smaller  than  the  dividend  ? 

Give  examples. 

52.  Does  the  quotient  always  give  the  number  of 
times  the  divisor  is  contained  in  the  dividend  ? 

LESSON    XIII. 

1.  40  is  |-  of  what  number  ? 

ANALYSIS. — 40  is  -|  x  the  number.  Hence  the  num- 
ber is  40  divided  ~by  |=  40  -=-  5  and  quotient  x  8. 

2.  27  is  T?g-  of  what  number  ?     J-  of  what  number  ? 

3.  32  is  |-  of  what  number?     f  of  what  number? 

4.  60  is  f-  of  what  number  ?    -ff-  of  what  number  ? 

5.  72  is  |-  of  what  number  ?    ^  of  what  number  ? 

6.  48  is  f  of  what  number?    -^  of  what  number? 

7.  49  is  T^  of  what  number?    J-  of  what  number? 

8.  54  is  T9T  of  what  number  ?    -f-  of  what  number  ? 

9.  63  is  -|  of  what  number  ?    |-  of  what  number  ? 

10.  20  is  f  of  what  number  ?    ^  of  what  number? 

11.  10  is  f  of  what  number? 

SOLUTION. — T/7 10  is  %  of  a  number  f  then  %  of  10,  or 
31,  i<?  £  of  the  same  number  ;  and  3^-  is  %  of  2  x  3-J-, 
or  6f . 

16  is  •$•  of  what  number? 

12.  15  is  f  of  what  number  ?    f  of  what  number  ? 

13.  50  is  f  of  what  number  ?     -y-  of  what  number  ? 

14.  9  is  |  of  what  number?    |£  of  what  number  ? 

15.  11  is  f  of  what  number  ?   f  of  what  number  ? 

16.  5£  is  i  of  what  jmmber  ?    f  of  what  number  ? 

17.  15f  is  f  of  what  number  ?    £  of  what  number  ? 

18.  7|  is  f  of  what  number?    TV  of  what  number? 

19.  f  is  f  of  what  number  ?    |  of  what  number  ? 


I 


108  FRACTIONS. 

20.  ^5-  is  f  of  what  number  ?    ^  of  what  number  ? 

21.  TT  *s  f  °f  what  number?     f  of  what  number? 

22.  A  man  paid  37  dollars  for  f  of  an  acre  of  land. 
What  was  the  cost  of  the  whole  acre  at  this  rate  ? 

23.  A  man  sold  his  horse  for  thirty  Collars,  which 
was  |-  of  what  it  cost  him.     What  was  the  cost  of  the 
horse  ? 

NOTE. — The  pupil  will  form  four  questions  like  the  above,  and  give 
the  solutions. 

LESSON    XIV. 

1.  20  is  ^  of  how  many  times  5  ? 
SOLUTION. — 20  is  -fa  of  50,  and  50  is  10  times  5. 

2.  72  is  y9^  of  how  many  times  16  ? 

3.  60  is  -j^-  of  how  many  times  18  ? 

4.  24  is  f  of  how  many  times  2? 

5.  48  is  -f  of  how  many  times  19? 

6.  12  is  |-  of  how  many  times  6  ? 

7.  11  is  f  of  how  many  times  5  ? 

8.  36  is  -^  of  how  many  times  11  ? 

9.  20  is  -^  of  5  times  what  number  ? 
SOLUTION. — 20  is  -^  of  50,  and  50  is  5  times  10. 

10.  16  is  f  of  6  times  what  number? 

11.  40  is  -fy  of  16  times  what  number? 

12.  18  is  -f-  of  i.0  times  what  number? 

13.  60  is  4  of  12  times  what  number? 

14.  20  is  how  many  tenths  of  10  times  5  ? 

15.  20  is  4  times  what  part  of  10  times  5  ? 

16.  What  number  is  -^  of  10  times  5  ? 

17.  48  is  ££  of  how  many  thirds  of  36  ? 

18.  54  is  T6T  of  how  many  times  -^  of  48  ? 

19.  55  is  -|  of  how  many  sixths  of  24? 

20.*  64  is  f-f  of  how  many  nineteenths  of  38  ? 

21.  9  is  |  of  how  many  sixths  of  18  ? 

22.  f  is  f  of  how  many  times  £  of  10  ? 

23.  8 J  is  f  of  how  many  times  J  of  24  ? 


FRACTIONS.  109 

24.  7i  is  |  of  how  many  times  -J-  of  10  ? 

25.  i  of  £  is  f  of  how  many  times  ^  of  100  ? 

26.  f  of  f  is  ^  of  how  many  times  -J-  of  ^  ? 

27.  A  man  bought  a  horse  for  $60,  which  was  •§-  of 
10  times  what  it  cost  to  keep  the  horse  one  month. 
How  much  did  it  cost  to  keep  the  horse  a  month  ? 

NOTE. — The  pupil  will  frame  three  questions  similar  to  question  27, 
and  give  the  solutions.  The  teacher  will  vary  some  of  the  questions, 
beginning  with  question  17,  after  the  manner  in  which  question  1  is 
varied  in  9,  14,  15,  and  16. 

LESSON    XV. 

1.  f  of  15  is  how  many  fifths  of  100  ? 
SOLUTION.— |  of  15  is  20.     20  is  \  of  100. 

2.  f  of  15  is  -|-  of  what  number  ? 

3.  f  of  what  number  is  -J-  of  100? 

4.  f-  of  48  is  how  many  fourths  of  12  ? 
SOLUTION.— f  of  48  is  30.    %  of  12  is  3.     30  is  10 

times  3,  or  10  times  £  of  12,  or  10  fourths  of  12. 

5.  ^  of  60  is  how  many  tenths  of  30  ? 

6.  -|  of  49  is  how  many  thirds  of  42  ? 

7.  f  of  9  is  how  many  fourths  of  12  ? 

8.  f  of  21  is  how  many  fourths  of  60  ? 

9.  |  of  16  is  how  many  sixths  of  36  ? 

10.  f  of  12  is  how  many  fifths  of  20  ? 

11.  f  of  48  is  how  many  tenths  of  40  ? 

12.  T5T  of  77  is  -J  of  what  number? 
SOLUTION.— fV  of  77  is  35.    35  is  -J  of  what  number  f 

13.  -jV  of  96  is  ^  of  what  number  ? 

14.  f  of  44  is  TV  of  what  number  t 

15.  |  of  63  is  %  of  what  number  ? 

16.  T\  of  72  is  f  of  what  number? 

17.  I  of  11  is  f  of  what  number  ? 

»18.  $  of  36  is  -&  of  how  many  times  5  ? 
19.  4  of  56  is  f  of  how  many  times  10  ? 

20.  |  of  54  is  TV  of  how  many  times  7? 

21.  f  of  27  is  f  of  how  many  times  14  ? 


110  FRACTIONS. 

22.  |  of  36  is  f;  of  how  many  times  10  ? 

23.  -£  of  12  is  -J-  of  how  many  times  5  ? 

24.  -^  of  ^  is  ^  of  how  many  times  ^  ?     How  much 
is  f  of  f  of  36  ? 

25.  f  of  f  of  60  is  f  of  what  number  ? 

SOLUTION. — |  of  60  w  40.    f  of  40  w  15.    15  is  4  o/ 

T      4  -L        9  *    J 

vial  number? 

26.  ^  of  -J  of  45  is  ||-  of  what  number? 

27.  f  of  f  of  35  is  \  of  what  number  ? 

28.  |  of  -^  of  100  is  f  of  what  number? 

29.  |  of  £  of  84  is  -J-  of  f  of  what  number? 

30.  f  of  |  of  48  is  f  of  t  of  what  number? 

31.  -|  of  36  is  -fa  of  4  times  what  number? 

32.  \  of  36  is  how  many  tenths  of  4  times  5  ? 

33.  f  of  63  is  -j^-  of  5  times  what  number  ? 

34.  f  of  80  is  |  of  12  times  what  number  ? 

35.  f  of  81  is  |f  of  8  times  what  number  ? 

36.  f  of  11  is  f  of  10  times  what  number? 

37.  f  of  f  of  24  is  |-  of  4  times  what  number  ? 

38.  |-  of  -fa  of  50  is  |f  of  5  times  what  number? 

39.  |-  of  -fa  of  50  is  how  many  tenths  of  5  times  6  ? 

40.  |-  of  fa  of  50  is  12  times  what  part  of  5  tynes  6  ? 

NOTE. — The  teacher  will  vary  the  above  questions  after  the  manner 
indicated  in  32,  39,  40.  The  pupil  will  form  four  questions  similar  to 
the  above,  and  give  the  solutions. 

LESSON    XVI. 

1.  |-  of  15  is  -fa  of  how  many  thirds'  of  21  ? 
SOLUTION.— f  of  15  is  12 ;  12  is  -faof2(};  £  of  21  is 
7;  20-^-7  gives  2f.     Hence,  iqflb  is -fa  of  2f  thirds 


2.  f  of  40  is  4  of  how  many  ninths  of  63  ? 

3.  -jSj-  of  77  is  |  of  how  many  tenths  of  100  ? 

4.  f  of  24  is  T9Q-  of  how  many  twentieths  of  80  ? 

5.  ±£-  of  30  is  ||-  of  how  many  thirds  of  36  ? 

6.  |  of  18  is  -I  of  how  many  twelfths  of  72  ? 


FRACTIONS.  Ill 


7.  £  of  9  is  ^  of  how  many  thirds  of 

8.  £  of  80  is  |  of  how  many  fifths  of  60  ? 

9.  T5T  of  39  is  -^  of  how  many  ninths  of  90  ? 

10.  |  of  -5%.  of  40  is  8  times  what  part  of  36  ? 

11.  |  of  f  of  48  is  10  times  what  part  of  24? 

12.  \  of  |  of  72  is  6  times  what  part  of  50  ? 

13.  TV  of  TV  of  90  is  TV  time  what  part  of  20  ? 

14.  4.  of  f  of  70  is  3  times  what  part  of  40  ? 

15.  f  of  48  is  -|  of  how  many  times  -f  of  45  ? 

16.  f  of  56  is  £  of  how  many  times  f  of  27  ?     54  is 
how  many  times  f  of  27. 

17.  f  of  64  is  -^Q  of  how  many  times  f  of  35  ? 

18.  f  of  63  is  f  of  how  many  times  f  of  40  ? 

19.  f  of  24  is  -f-  of  how  many  times  i  of  20  ? 

20.  |  of  60  is  T^  of  how  many  times  f  of  24  ? 

21.  f  of  %  of  96  is  TV  of  how  many  times  f  of  12  ? 

22.  |  of  81  is  |  of  how  many  times  %  of  f  of  40? 
How  much  is  ^  of  f  of  40  ?     -|  of  81.  is  f  of  what 
number  ? 

23.  f  of  £  of  60  is  ^  of  how  many  times  f  of  f  of 
24? 

24.  ^  of  T6g-  of  100  is  f  of  f  of  how  many  times  -|  of 
|  of  30  ?     I-  of  T<V  of  100  is  how  much  ?     12  is  |  of  f  of 
what  number  ?     How  much  is  f  of  |-  of  30  ? 

25.  f  of  f  of  72  is  f  of  f  of  how  many  times  ^  of  ^ 
of  54? 

NOTE.  —  The  pupil  will  form  three  questions  similar  to  the  above,  and 
give  the  answers. 

LESSON    XVII. 

1.  A  farmer  having  70  sheep  sold  ^  of  them  at  one 
time,  and  ^  of  them  at  another.     How  many  of  them 
had  he  left  ? 

2.  A  woman  is  50  years  old,  and  her  age  is  |  of  the 
age  of  her  husband.     How  old  is  he  ? 

3.  I  bought  a  horse  for  $100  ;  I  paid  f  as  much  for 


112  FRACTIONS. 

« 

an  ox,  and  I-  as  mucli  for  a  colt.     How  much  did  I 
pay  for  all  three  ? 

4.  A  man  bought  40  dozen  eggs  for  400  cents.     He 
sold  f  of  them  for  15  cents  a  dozen,  and  the  remainder 
at  10  cents  a  dozen.     How  much  did  he  gain  ? 

5.  Iff  of  a  barrel  of  flour  cost  12  dollars,  what  will 
be  the  cost  of  a  barrel? 

6.  A  farm  contains  60  acres  of  wood  land,  and  -f| 
of  the  wood  land  is  4  of  the  cleared  land.     How  much 
cleared  land  in  the  larm  ? 

7.  A  person  owned  in  a  town  17  lots,  each  lot  con- 
taining -J-  of  an  acre.     How  many  acres  did  he  own  ? 

8.  From  a  lot  of  5  acres  of  land,  I  sold  %  an  acre  to 
one  man,  f  of  an  acre  to  another,  and  -^  to  a  third. 
How  many  acres  did  I  have  left  ? 

9.  I  paid  out  of  a  sum  of  $100,  at  different  times, 
20^-  dollars,  30f  dollars,  40|-  dollars.     How  much  did 
I  have  left  ? 

10.  Frank  gave  f  of  an  orange,  J  of  an  orange,  and 
•f  of  an  orange  to  one  boy ;  J  of  an  orange,  f  of  an 
orange,  and  -£%  of  an  orange  to  another  boy.     How 
much  more  did  he  give   to  the  first  boy  than  the 
second  ? 

11.  What  is  the  cost  of  4  yards  of  tape  at  -J  of  a 
dollar  a  yard  ? 

12.  What  is  the  cost  of  10  yards  at  -^  of  a  dollar  a 
yard? 

13.  What  is  the  cost  of  9  barrels  of  flour  at  6f  dol- 
lars a  barrel  ? 

14.  A  man  paid  12  dollars  for  3  yards  of  cloth,  and 
sold  it  at  5f  dollars  a  yard.    How  much  did  he  gain  ? 

15.  What  is  the  difference  between  5|  times  10  and 
3  times  8f  ? 

16.  5  times  7  and  -f-  of  7  is  how  many  times  10  ? 

17.  If  a  bushel  of  potatoes  is  worth  ^  a  dollar,  how 
touch  is  -J-  of  a  bushel  worth  ?     f  of  a  bushel  ?     f  of  a 
bushel  ?    -j^-  of  a  bushel  ? 


FRACTIONS.  113 

18.  If  a  yard  of  cloth  is  worth  ^  of  a  dollar,  what 
is  J  of  a  yard  worth  ?     f  of  a  yard  ?     ^  of  a  yard  ? 

19.  What  is  the  cost  of  2J-  yards  of  cloth  at  f  of  a 
dollar  a  yard  ?     At  1|-  dollar  a  yard  ?     At  5-f  dollars 
a  yard  ?    At  6-|  dollars  a  yard  ?    At  3£  dollars  a  yard? 

20.  I  bought  a  horse  for  82^  dollars,  and  sold  him  for 
%  of  that  sum.     What  did  I  lose  on  him  ? 

21.  If  a  man  can  mow  l^-  acre  of  grass  in  one  day, 
how  much  can  he  mow  in  5|-  days  ? 

22.  What  will  -^  of  %  of  70  gallons  of  wine  cost  at 
If  dollar  a  gallon  I 

23.  At  -|  of  a  dollar  a  bushel,  how  many  bushels  of 
meal  can  be  bought  for  five  dollars  ?  for  6  dollars  ?  10 
dollars  ?  20  dollars  ? 

24.  If  a  horse  eats  J-  of  a  bushel  of  oats  a  day,  how 
•long  will  15  bushels  of  oats  last  him?     How  long  5 

bushels  ?  7  bushels  ? 

25.  If  12  pounds  of  sugar  cost  -f  of  a  dollar,  what 
is  the  cost  of  1  pound  ? 

26.  At  f  of  a  dollar  a  yard,  how  much  cloth  can  be 
bought  for  f  of  a  dollar  ?  for  f  of  a  dollar  ? 

27.  If  7  bushels  of  wheat  cost  8^-  dollars,  what  i»  the 
cost  of  one  bushel  ? 

28.  In  y9^  of  an  acre  of  land,  how  many  building 
lots  of  4-  of  an  acre  each,  and  how  much  over  ? 

29.  If  a  horse  eats  f  of  a  bushel  of  oats  a  day,  how 
long  will  |~J-  of  a  bushel  last  him  ? 

30.  If  f  of  a  pound  of  tea  costs  1  dollar,  what  is  the 
cost  of  f  of  a  pound  ? 

31.  If  f  of  a  bushel  of  barley  be  given  for  f  of  a 
bushel  of  corn,  and  the  corn  costs  f  of  a  dollar  a 
bushel,  what  is  the  cost  of  a  bushel  of  barley? 

32.  In  an  orchard  |-  of  the  trees  are  apple-trees, 
^  peach-trees,  J  pear-trees,  and  the  remaining  11  are 
plum-trees.     How  many  trees  in  the  orchard? 

33.  What  number  added  to  4  times  f  of  21  will 
make  52  ? 


PROPORTION. 

34.  24  is  f  of  some  number.    What  is  ^  of  the  same 
number  ? 

35.  How  long  will  it  take  two  wood-choppers  to 
cut  14:  cords  of  wood,  if  one  of  them  cuts  f  of  a  cord  a 
day,  and  the  other  cuts  1  cord  ? 

36.  Several  workmen  received  together  6f  dollars, 
and  the  wages  of  each  one  was  -f-  of  a  dollar.     How 
many  workmen  were  there  ? 

37.  A  man  plants  a  bed  of  36  square  feet  in  cab- 
bage, and  gives  to  each  cabbage  f  of  a  square  foot. 
How  many  cabbages  can  he  plant  in  the  bed  ? 

38.  How  many  steps  in  a  staircase  39  feet  high, 
when  each  step  is  l^  foot  high  ? 

39.  A  yard  of  a  certain  cloth,  4  quarters  wide,  is 
worth  3  dollars.     What  will  1  yard  of  the  same  cloth, 
5  qn  arters  wide,  be  worth  ? 

NOTE. — The  pupil  will  frame  four  questions  similar  to  the  above, 
and  give  the  answers. 

LESSON   XVIII. 

PROPORTION. 

1.  What  is  the  difference  between  6  and  2?     Give 
two  other  numbers  which  have  the  same  difference 
between  them. 

2.  If  we  take  8  and  4  as  the  two  numbers,  we  will 
have  6  less  2  equal  to  8  less  4.     Now  if  we  add  the 
greater  of  the  first  pair  of  numbers  to  the  less  of  the 
second  pair,  and  the  less  of  the  first  to  the  greater  of 
the  first,  what  are  the  results?     Try  the  same  with 
any  pair  whose  difference  is  4. 

3.  Find  5  pairs  of  numbers,  the  difference  between 
the  two  forming  each  pair  being  5. 

4.  What  number  is  as  much  greater  than  10  as  12 
is  greater  than  7  ?     How  is  this  number  found  ? 

ANS. — By  adding  the  difference  between  12  and  7  to 
10. 


PROPORTION.  115 

5.  What  is  the  quotient  of  6-^-2?     Give  two  other 
numbers  which  have  the  same  quotient.     If  we  take 
the  numbers  12-^4,  we  will  have  6-f-2  equal  to  12-^-4. 
If,  now,  we  multiply  the  greater  of  the  first  pair  of 
numbers  by  the  less  of  the  second  pair,  and  the  less 
of  the  first  pair  by  the  greater  of  the  second,  what  are 
the  results  ?     Try  the  same  with  another  pair  of  num- 
bers whose  quotient  is  3. 

6.  Find  5  pairs  of  numbers,  the  quotient  of  the 
numbers  in  each  pair  being  5. 

7.  What  number  is  contained  in  15  as  often  as  4  is 
contained  in  20?  or  20-^4  is  equal  to  15-7- what  num- 
ber ?     How  is  this  number  found  ? 

ANS. —  We  divide  15  by  the  quotient  of  20  divided 
1y  4. 

'8.  12  -r-  3  is  equal  to  16  ~-  what  number?  12  —  4 
=  15-7-  what  number?  10  -f-  2  =  20  -=-  what  num- 
ber? 

9.  20  -r-  4  is  called  the  ratio  of  20  to  4.     The  line 
between  the  two  dots  is  left  out  for  convenience ;  thus, 
20  :  4,  which  is  read  20  to  4. 

10.  The  ratio  of  20  to  4  is  -2/,  or  5. 

So,  12  ---  3  =12  :3=-V2-  — 4;  3:  12— •&•—{•; 
5  :  7  -  f;  1  :  3±  -  I  +  J-  -  1  x  f  -  f  ;  3J  :  1= 
jni  .  l  _  ^ 

What  is  the  ratio  of  10  :  4?  of  5  :  10 ?  of  3  :  9 ?  of 
9:3?  of  5  :  11  ?  of  11  :  5  ?  of  8-J- :  1J  ?  of  f  :  f  ?  of  2£ 
:  5?  of  5  :  2|? 

11.  Find  6  pairs  of  numbers  that  have  the  same 
ratio  as  10  :  5. 

12.  4  :  12  =-  13  :  what  number? 

SOLUTION. — 4  :  12  =  ^.     The  required  number  is, 
en,  13  -=-  4  =  13  x  3,  or  39.     Or,  since  12  is  3  times 
the  required  number  must  be  3  x   13,  or  39. 

13.  1:5  =  7:?    9:7  =  14  :?   6:9=8:?    2:5 
7:?    3:7  =  6:?  —  3:7  =  6:  14  is  called  a  PRO- 

ORTION. 


116  PROPORTION. 

14.  What  is  the  ratio  of  one  number  to  another? 
(See  Q.  9.) 

When  are  .four  numbers  said  to  be  in  proportion  ? 
Ans.  When  the  ratio  of  the  1st  and*%d  =  the  ratio  of 
the  3d  and  4tk. 

How  do  you  find  the  fourth  term  of  a  proportion 
when  three  terms  are  known  ?  (See  Q.  12.) 

15.  4  :  10  —  11  :  ?    5  :  J  ==  40  :  ?    6  :  4  =  7J  :  ? 
1  :  2f  ==  4£  :  ?    (1  :  2|  —  f .    The  required  number  is, 
then,  4|--t  =  fxf  =  ^  =  12).     *  :  f  =  |  :  ? 

16.  12  is  f  of  what  number  ?     f  :  1  =  12  :  what 
number  ?    18  is  f  of  what  number  ?    6:7  =  18:  what 
number?     f  :  1  =  18  :  what  number? 

17.  6  workmen  in  2^-  days  will  do  as  much  as  10 
workmen  in  how  many  days  ? 

SOLUTION. — 6  :  10  =  f .  The  number  of  days  re- 
quired is,  then,  2i~£  =  J--f-£=fxf  =  -V5-, 
or  4-J-. 

6  men  in  12 \  days  do  as  much  as  15  men  in         days  ? 

12  men  in  6f  days  do  as  much  as  8  men  in         days  ? 

6  men  in  7|-  days  do  as  much  as  4  men  in         days  ? 

18.  20  days'  work  for  six  men  is  10  days'  work  for 
how  many  ? 

3f  days'  work  for  14  men  is  7|-  days'  work  for 
men? 

19.  7  days'  rations  for  6  men  is  how  many  days' 
rations  for  4  men  ?  , 

6f  days'  rations  for  5  men  is  how  many  days'  rations 
for  7  men  ? 

3f  days'  rations  for  14  men  is  how  many  days'  ra- 
tions for  7  men  ? 

2^  days'  rations  for  8  men  is  how  many  days'  rations 
for  0  men  ? 

20.  3  men  do  a  piece  of  work  in  5  days ;  how  long 
will  it  take  7  men  to  do  the  same  work  ? 

5  men  in  6  days  do  as  much  as  11  men.  in 
days? 


PROPORTION.  117 

21.  3  days'  rations  for  11  men  are  how  many  days' 
rations  for  13  men*? 

22.  3  men  can  'do  a  piece  of  work  in  5  days ;  how 
long  will  it  take  7  men  to  do  twice  the  work  ?   3  times 
the  work  ?  4  times  the  work  ?  |-  the  work  ? 

23.  If  5  men  build  a  wall  50  feet  long  in  6  days,  in 
what  time  will  9  men  build  a  similar  wall  150  feet  long? 

24.  3  days'  rations  for  12  men  are  16  days'  half-rations 
for  how  many  men  ? 

25.  If  3  horses  eat  8  bushels  of  oats  in  2  weeks,  how 
long  will  40  bushels  last  them?     How  long  will  40 
bushels  last  5  horses  at  the  same  rate  ? 

26.  How  many  yards  of  cloth,  3  quarters  wide,  are 
equal  to  7  yards  of  the  same  cloth  5  quarters  wide  ? 

27.  How  many  yards  of  carpeting,  1  yard  wide,  are 
equal  to  20  yards  £  of  a  yard  wide  ? 

28.  A  workman  laboring  11  hours  a  day  does  a 
certain  piece  of  work  in  5  days ;  in  how  many  days 
will  he  do  the  same  work,  laboring  5  hours  a  day  ? 

29.  A  traveller  has  enough  money  to  last  him  32 
days,  at  J  of  a  guinea  a  day ;  how  many  days  will  it 
last  him  if  he  limits  his  expenses  to  f  of  a  guinea  a 
day? 

30.  A  man  bought  6  barrels  of  flour  at  7  dollars  a 
barrel;  at  what  price  per  barrel  must  he  sell  it  in 
order  to  gain  11  dollars  ? 

31.  A  man  being  asked  how  much  he  had  given  for 
each  of  two  horses,  stated  that  he  gave  $60  for  the 
first,  and  f  of  the  price  of  the  first  was  J-  of  •§•  of  the 
price   of  the   second.     What   was   the   price   of  the 
second  ? 

32.  A  man  can  do  a  piece  of  work  in  4f  days,  if  he 
labor  10  hours  a  day ;   how  many  days  will  it  take 
him  to  do  the  work,  laboring  7  hours  a  day  ?   • 

33.  If  5  bushels  of  oats  cost  2f  dollars,  what  will  7 
bushels  cost? 

34.  If  10  yards  of  cloth  cost  45  dollars,  what  will 


118  PROPORTION. 

be  the  cost  of  6  yards  of  cloth  worth  twice  as  much 
per  yard  ? 

35.  If  6  yards  of  cloth  cost  2|-  dollars,  what  cost  11 
yards  ? 

36.  If  5  yards  of  cloth  cost  7  dollars,  what  cost  13 
yards  ? 

37.  If  8  bushels  of  meal  cost  10  dollars,  how  much 
can  be  bought  for  12  dollars  ? 

38.  If  9  barrels  of  flour  cost  58£  dollars,  how  many 
can  be  bought  for  75  dollars  ? 

39.  If  10  acres  of  land  cost  300  dollars,  what  is  the 
cost  of  50  acres  ? 

40.  At  the  rate  of  10  acres  for  500  dollars,  how 
much  land  can  be  bought  for  3000  dollars  ? 

41.  A  railway  train  travels  at  the  rate  of  90  miles 
in  3  hours ;  how  many  miles  will  it  go  in  10  hours  ? 

42.  If  a  railway  train  travels  at  the  rate  of  100  miles 
in  5  hours,  how  many  hours  will  it  require  to  go  320 
miles  ? 

NOTE. — In  this  and  all  the  succeeding  lessons  the  pupils  should  be 
required  to  form  questions  for  themselves  and  each  other. 


DENOMINATE   NUMBEBS.  119 

CHAPTER    YIII. 

DENOMINATE    NUMBERS. 

LESSON    I. 
Table  of  Long  Measure. 

This  table  is  used  in  measuring  distances. 

The  smallest  units  are  the  inch  and  its  subdivisions. 

1  Inch. 


12  inches  (in.)  make  1  foot — marked  ft, 

3  feet  make  one  yard — marked  yd. 

5|-  yards  make  1  rod  or  pole — marked  rd. 

40  rods  make  1  furlong — marked  fur. 

8  furlongs  make  1  mile — marked  m, 

3  miles  make  1  league — marked  lea. 

69J  miles  (nearly)  }  make  1  degree — marked  deg. — 

60  nautical  miles    \  of  the  circumference  of  the  earth. 

360  degrees  make  1  circumference — marked  circ, 

1.  How  many  inches  in  2  feet  ?  3  feet  ?  5  feet  ?  in 
10  feet?  in  \  of  a  foot?  £  of  a  foot?  £  of  a  foot?  \  of 
a  foot  ?  f  of  a  foot  ?  in  f  of  a  foot  ? 

2.  How  many  feet  in  24  inches?    36  inches?    30 
inches?  in  15  inches?  in  17  inches?  in  44  inches? 

3.  How  many  feet  in  2  yards  ?  11  yards  ?  3£  yards  I 
4|  yards  ?  in  ^  of  a  yard  ?  in  f  of  a  yard  ?   What  part 
of  a 'yard  is  1  foot?     What  part  of  a  yard  is  1  inch? 

4.  How  many  yards  in  6  feet?    8  feet?    9  feet?    12 
feet?  13  feet?  25  feet? 

5.  How  many  yards  in  3  rods  ?  in  5  rods  ?  in  10 
rods  ?  in  3£  rods  ?  in  40  rods  ? 

6.  How  many  rods  in  2  furlongs  ?  3  furlongs  ?   in  6 


120  DENOMINATE   NUMBERS. 

furlongs  ?  in  one  mile  ?  What  part  of  a  mile  is  a  fur- 
long? What  part  of  a  mile  is  one  rod?  How  many 
furlongs  in  40  rods?  in  50  rods?  in  120  rods?  in  150 
rods? 

7.  How  many  furlongs  in  2  miles?  Smiles?  4^  miles? 
f  of  a  mile  how  many  furlongs  ?   -^  of  a  furlong  is  what 
part  of  a  mile?     How  many  miles  in  16  furlongs?  in 
24  furlongs  ?  in  25  furlongs  ?  in  40  furlongs  ?  in  45 
furlongs  ?     How  many  yards  in  -J  of  a  mile  ?  in  J  of  a 
mile  ?  in  ^  of  a  mile  ? 

8.  How  many  miles  in  3  leagues  ?   in  4  leagues  ?  in 
10  leagues  ?     A  mile  is  what  part  of  a  league  ?    f  of  a 
league  are  how  many  miles  ? 

9.  How  many  leagues  in  9  miles  ?  in  11  miles  ?  in 
24  miles  ? 

10.  If  a  man  travels  a  mile  in  ten  minutes,  in  how 
many  minutes  will  he  travel  1  league?    3  leagues  ?    4 
leagues  ? 

11.  If  a  ship  sails  20  leagues  a  day,  how  long  will 
it  take  her  to  sail  600  miles?  300  miles?  3000  miles ? 

12.  How  many  inches  in  3  yds.  2  ft.  4  in.  ?  in  2  yds. 
1  ft.  5  in.  ? 


Table  of  Cloth  Measure. 

This  measure  is  used  in  measuring  cloth  and  other 
articles  sold  by  the  yard.  The  yard  and  inch  are  the 
same  in  this  as  in  Long  Measure. 

• 

2J  inches  (in.)  make  1  nail — marked  na. 

4  nails  make  1  quarter  of  a  yard — marked  qr. 

4  quarters  make  1  yard — marked  yd. 

5  quarters  make  1  ell  English — marked  E.  E. 

1.  How  many  inches  in  J  of  a  yard  ?  How  many 
in  |-  a  yard  ?  in  f  of  a  yard  ?  How  many  inches  in  an 

idly 


DENOMINATE    NUMBERS. 


121 


2.  How  many  quarters  in  12  nails  ?  13  nails  ?  in  24 
nails  ? 

3.  How  many  yards  in  6  quarters  ?   in  20  quarters  ? 
How  many  nails  in  3  yds.  3  qrs.  3  nails  ?     How  many 
inches  in  3  qrs.  2  na.  4  in.  ? 

4.  If  4:  nails  of  cloth  cost  20  cents,  what  is  the  cost 
of  5  yards  ?     What  part  of  a  yard  is  one  nail  ?     What 
part  of  an  ell  is  a  quarter  ? 

5.  What  is  the  cost  of  12  yards  of  cloth  at  5  dollars 
an  ell  ?     f  of  a  yard  is  what  part  of  an  ell .?  what  part 
of  a  rod?    f  of  a  rod  is  what  part  of  a  furlong?     -^  of 
a  yard  is  what  part  of  a  foot  ?     TV  of  a  foot  is  how 
many  inches  ? 

LESSON    II. 
Square  Measure. 

Square   Measure   is   used   for  measuring   surfaces. 
The  square  inch  is  the  smallest  unit  in  the  table. 


Square  Inch. 


TABLE. 

square  inches  (sq.  in.)  make  one  square  foot 
marked  sq.  ft. 

9  square  feet  make  1  square  yard — marked  sq.  yd. 

30J  square  yards  make  1  square  rod  or  perch 
marked  P. 

40  square  rods  make  1  rood — marked  E. 

4  roods  make  1  acre — marked  A. 

640  acres  make  1  square  mile — marked  sq.  m. 

1.  How  many  square  feet  in  2  square  yards?  in  2 


122 


DENOMINATE   NUMBERS. 


of  a  square  foot  is 
of  a  square  yard  is 


or 


square  yards  ?  in  5  square  yards  ;  in  3£  square  yards  ? 
How  many  square  yards  in  36  square  feet?   in   81 
square  feet  ?  in  44  square  feet  ? 
what  part  of  a  square  yard  ? 
how  man}'  square  feet  ? 

2.  How  many  square  yards  in  3  square  rods 
perches?  in  4  perches?  in  10  perches?     How  many 
square  rods  in  3  roods  ?  in  \  of  a  rood  ?  in  f  of  a  rood? 
in  3^-  roods  ?     What  part  of  a  rood  is  a  square  rod  ? 

3.  How  many  acres  in  10  roods?    16  roods?    40 
roods  ?     What  part  of  an  acre  is  1  rood  ?   ^  of  a  rood  ? 
-jig-  of  a  rood  ?     How  many  roods  in  f  of  an  acre  ?     20 
square  rods  is  what  part  of  an  acre  ? 

4.  At  5  dollars  a  square  rod,  what  is  the  cost  of  2 
roods  and  5  sq.  rods  of  land  ?     1  A.  2  R.  15  P.  are 
how  many  perches  ? 

5.  At  2f  dollars  a  rood,  what  is  the  cost  pf  2  acres 
of  land  ? 

Solid  or  Cubic  Measure. 

This  measure  is  used  in  measuring  timber  and  stoneT 
and  gives  the  unit  for  Liquid  and  Dry  Measure,  and 
measures  weights  and  coins. 


Cubic  Inch. 

TABLE. 

1728  cubic  inches  (cu.  in.)  make  1  cubic   foot — 
marked  cu.  ft. 

27  cubic  feet  make  1  cubic  yard — marked  cu.  yd. 
16  cubic  feet  make  1  cord  foot — marked  co.  ft. 


DENOMINATE    NUMBERS.  123 

8  cord  feet,  or  128  cubic  feet,  make  1  cord  of  wood 
— marked  0. 

1.  How  many  cubic  feet  in  2  cubic  yards?    in  3 
cubic  yards?  in  -J-  of  a  cubic  yard?  in  ^  of  a  cubic 
yard  ?  in  -^  of  a  cubic  yard  ? 

2.  How  many  cubic  yards  in  27  cubic  feet?  in  36 
cubic  feet?  in  54  cubic  feet?  in  57  cubic  feet?     What 
part  of  a  cubic  yard  is  1  cubic  foot  ?  3  cubic  feet  ?  9 
cubic  feet?     How  many  cubic  feet  in  f  of  a  cubic 
yard? 

3.  How  many  cord  feet  in  2  cords  of  wood  ?  in  2^- 
cords  of  wood  ?  in  If  cord  of  wood  ?     |-  of  a  cord  is  - 
how  many  cord  feet  ? 

4.  How  many  feet  of  plank,  1  inch  thick,  in  50  feet 
of  plank  2  inches  thick  ?  4|-  inches  thick  ? 

LESSON     III. 
Liquid  or  Wine  Measure. 

This  measure  is  used  in  measuring  all  kinds  of 
liquids.*  The  wine  pint-cup  will  hold  28^-  cubic 
inches. 

TABLE. 

4:  gills  (gi.)  make  1  pint — marked  pt. 
2  pints  make  1  quart — marked  qt. 
4  quarts  make  1  gallon — marked  gal. 
63  gallons  make  1  hogshead — marked  hhd. 
2  hogsheads  make  1  pipe — marked  pi. 
2  pipes  make  1  tun — marked  tun. 


I 


1.  How  many  gills  in  2  pints  ?  3  pints  ?  |-  of  a  pint? 
of  a  pint?  in  2^  pints?   in  6f  pints?     How  many 
pints  in  8  gills?    9  gills?    15  gills?    60  gills?     What 

Kart  of  a  pint  is  1  gill  ?  2  gills  ?  3  gills  ? 
2.  How  many  pints  in  5  quarts  ?    3  quarts  ?    10-J- 
quarts?   2^-  quarts?   in  12  quarts?     How  many  gills 
in  2  quarts?     What  part  of  a  quart  is  1  pint?     How 
many  quarts  in  12  pints?  in  13  pints?  50  pints? 


124  DENOMINATE   NUMBERS. 

3.  How  many  quarts  in  3  gallons  ?    2£  gallons  ?    5f 
gallons?     What  part  of  a  gallon  is  a  quart?     How 
many  gallons  in  2  hogsheads  ?     9  gallons  is  what  part 
of  a  hogshead?    21  gallons  is  what  part  of  a  hogshead? 

4.  If  3  quarts  of  oil  cost  60  cents,  what  is  the  cost 
of  5  gallons  and  2  quarts  ? 

5.  How  many  quart  bottles  can  you  fill   from  a 
hogshead  of  wine  ?     How  many  gallons  in  a  pipe  ? 

Dry  Measure. 

This  measure  is  used  in  measuring  grain,  fruit,  coal, 
salt,  &c.     A  pint-cup  of  this  measure  contains 
cubic  inches. 

TABLE. 

2  pints  (pt.)  make  1  quart — marked  qt. 
8  quarts  make  1  peck — marked  pk. 
4  pecks  make  1  bushel — marked  bu. 

1.  How  many  cubic  inches  larger  is  the  fynt,  Dry 
Measure,  than  the  pint,  Liquid  Measure  ?     How  much 
larger  is  the   quart,  Dry  Measure,  than  the   quart, 
Liquid  Measure? 

2.  How  many  quarts  in  2  pecks  ?   in  4|-  pecks  ?   in 
3f  pecks  ?  ^n  ^  peck  ?  f  of  a  peck  ?  -f-  of  a  peck  ?   How 
many  pecks  in  16  quarts?    18  quarts?    21  quarts?   40 
quarts  ?     What  part  of  a  peck  is  a  pint  ? 

3.  How  many  pecks  in  4  bushels  ?  in  ^  of  a  bushel  ? 
in  5£  bushels  ?   in  10  bushels  ?  in  100  bushels  ?     How 
many  bushels  in  12  pecks?  in  15  pecks?  in  32  pecks? 
in  45  pecks? 

4.  What  is  the  cost  of  1  bushel  of  corn  at  5  cents  a 
quart-?    What  of  2  quarts  of  corn  at  60  cents  a  bushel  ? 

5.  How  many  quarts  in  4  bushels  ?  in  3£  bushels  ? 
in  2  bu.  3  pk.  3  qt.  ? 

6.  If  2  pints  of  meal  cost  5  cents,  what  is  the  cost 
of  a  bushel  ?     What  part  of  a  bushel  is  a  pint  ? 


DENOMINATE   NUMBERS.  125 

LESSON    IV. 
Troy  Weight. 

Troy  Weight  is  used  in  weighing  gold,  silver,  and 
jewels.  The  Troy  pound  is  a  piece  of  brass  or  other 
metal,  which  weighs  as  much  as  22T8T  cubic  inches  of 
water,  at  a  certain  temperature. 

TABLE. 

24  grains  (gr.)  make  1  pennyweight — marked  pwt. 
20  pennyweights  make  1  ounce — marked  oz. 
12  ounces  make  1  pound — marked  Ib. 

1.  In  1  pennyweight  how  many  grains  ?  in  2  ?  in  3  ? 
in  4£  ?  in  5£  ? 

2.  How  many  pennyweights  in  48  grains  ?  in  72  ? 
in  75? 

3.  In  1  pound  how  many  ounces  ?  in  2  ?  in  4  ?  in  5  ? 
in  |?  in  2f? 

4.  In  24  oz.  how  many  Ibs.  ?  in  36  ?  48  ?  28  ?  39  ? 

Apothecaries'  Weight. 

The  Troy  pound  is  sometimes  divided  differently, 
to  weigh  medicines  in  mixing  them.  The  pound, 
ounce,  and  grain  are  the  same  in  Troy  and  Apotheca- 
ries' weight. 

20  grains  (gr.)  make  1  scruple — marked  sc.  or  3. 
3  scruples  make  1  dram — marked  dr.  or  3. 
8  drams  make  1  ounce — marked  oz.  or    §• 
12  ounces  make  1  pound — marked  Ib.  or  ft>. 

1.  In  1  scruple  how  many  grains?  in  2?  in  3?  in  \ 
of  a  scruple  ?  in  2|-  scruples  ? 

2.  In  2  drams  how  many  scruples  ?  in  3  ?   in  4  ?   in 
^  of  a  dram  ?  in  2-f  drams  ? 


126  DENOMINATE    NUMBERS. 

t3?_4?    I 

many 


3.  In  2  ounces  how  many  drams  ?  in  3  ?  4  ?  2f  ? 

4.  In  y^  of  a  pound  how  many  grains  ?     How  m£ 
grains  in  4  drams,  3  scruples,  5  grains  ? 


Avoirdupois  Weight. 

This  weight  is  used  in  all  ordinary  weighing.  The 
Avoirdupois  pound  is  heavier  than  the  Troy  pound, 
144  Avoirdupois  pounds  being  equal  to  175  Troy 
pounds. 

TABLE. 

16  drams  make  1  ounce — marked  oz. 

16  ounces  make  1  pound — marked  Ib. 

25  pounds  make  1  quarter — marked  qr. 

4  quarters,  or  100  Ibs.,  make  1  hundred  weight — 
marked  cwt. 

20  hundred  weight  make  1  ton,  marked  T. 

1.  In  2  pounds  how  many  ounces  ?   in  3  ?   in  4  ?    in 
%  of  a  pound  ?  in  2f  ? 

2.  In  32  ounces  how  many  pounds  ?  in  48?  64?  80? 
82? 

3.  How  many  pounds  in  2  quarters?  in  3?  in  4^? 

4.  How  many  quarters  in  50  pounds  ?  100  pounds  ? 
200  pounds  ?  76  pounds  ? 

5.  How  many  cwt.  in  8  quarters  ?  in  16  ?  in  25  ?  in 
33? 

6.  How  many  cwt.  in  2  tons?  4  tons?  \  a  ton  ?  -fa 
of  a  ton  ?  in  f  of  a  ton  ?  in  5£  tons  ?     What  part  of  a 
ton  is  1  cwt.  ? 

7.  How  many  tons  in  40  cwt.  ?  in  80  ?  in  60  ?  in  45  ? 

8.  What  is  the  cost  of  4|-  cwt.  of  sugar,  at  9  cents  a 
pound?  at  10  cents? 

9.  What  is  the  cost  of  5£  tons  of  hay,  at  20  dollars 
a  ton  ? 

10.  How  much  will  2  cwt.  2  qrs.  of  sugar  cost  at  10 
cents  a  pound  ? 

11.  Bought  3  tons  and  10  cwt,  of  hay  at  12  dollars 
a  ton  ?    What  was  the  cost  of  it  ? 


DENOMINATE   NUMBERS. 


127 


LESSON   Y. 
United  States  Money. 

The  units   are  coins   of  gold,  silver,  copper,   and 
nickel,  of  fixed  weight,  size,  and  shape. 

TABLE. 


Gold. 


Gold. 


Nickel. 


Silver. 


Silver. 


10  mills  make  one  cent. — marked  c.  » 

10  cents  make  1  dime — marked  d. 
10  dimes  make  1  dollar — marked  $. 
10  dollars  make  1  eagle — marked  E. 

1.  How  many  mills  in  2  cents?  9  cents?    2}  cents? 
How  many  cents  in  50  mills?  40  mills?  25  mills? 

2.  How  many  cents  in  2  dimes  ?   in  3  ?   in  4^  ?   in 
5|?     How  many  dimes  in  20  cents?   25  cents?    50 
cents  ?  75  cents  ? 


128  DENOMINATE    NUMBERS. 

3.  How  many  dimes  in  2  dollars  ?   4£  dollars  ? 
dollars?  6^  dollars?     How  many  dollars  in  20  dimes"? 
40  dimes?  45  dimes? 

4.  How  many  dollars  in  2  eagles?   in  a  half-eagle? 
in  3^  eagles?  in  5J  eagles?     How  many  eagles  in  20 
dollars  ?  35  dollars  ?  42  dollars  ? 

5  What  part  of  a  dollar  is  1  cent  ?  is  1  mill  ?  is  50 
cents  ?  33|  cents  ?  25  cents  ?  20  cents  ?  1  dime  ?  16| 
cents?  12^  cents? 

6.  What  part  of  an  eagle  is  1  djme  ?  5  dimes  ? 

7.  How  many  cents  in  |-  of  a  dollar  ?  in  J  ?  in  £  ?  in 
-£?  in  |?  in  |?  -jV?  ^-?  ^-?  ^-  of  a  dollar?  £  of  a 
dollar? 

8.  Bought  six  barrels  of  flour  at  $9.33^  cents  a  bar- 
rel ;  what  was  the  cost  of  the  whole  ? 

9.  Paid  6J  cents  a  pound  for  48  pounds  of  nails ; 
how  many  dollars  did  the  whole  cost  ? 

10.  At  37J  cents  a  yard,  what  is  the  cost  of  6  yards 
of  ribbon  ? 

11.  If  6  pounds  of  sugar  can  be  bought  for  60  cents, 
how  much  can  be  bought  for  an  eagle? 

12.  What  is  the  cost  of  100  articles  at  1  cent  each  ? 
2  cents?  3  cents?  12  cents? 

Table  of  English  Money. 

4  farthings  (far.)  make  1  penny — marked  d. 

12  pence  make  1  shilling — marked  s. 
.20  shillings  make  1  pound — marked  £. 

The  coins  of  this  table  are  gold,  silver,  and  copper. 
A  sovereign  is  a  gold  coin  =  1  pound  or  20  shillings. 
The  guinea  is  21  shillings.  A  pound  is  4  dollars  and 
84  cents. 

1.  How  many  farthings  in  2  pence  ?  in  3  pence  ?  in 
6J  pence  ?  ^  a  penny  ?  f  of  a  penny  ?  in  4f  pence  ? 

2.  How  many  pence  in  8  farthings  ?   in  10  ?   in  20  ? 
in  60? 


DENOMINATE    NUMBEKS.  129 

3.  How  many  pence  in  2  shillings  ?  in  -J-  of  a  shil- 
ling  ?  in  \  of  a  shilling  ?  in  £  of  a  shilling  ?  -jj-  of  a 
shilling  ?  in  5s.  6d.  ?  in  10s.  2d.  ?  in  10s.  6d.  ?  in  2s. 
8<U 

4.  How  many  shillings  in  24  pence  ?  48  pence  ?  32 
pence  ?  100  pence  ? 

5.  How  many  shillings  in  £2?  in  £-£?  in  £-J-?  in 
£4?  in£TV?  in  £2?  £2£?  £5?  in  £7?  in  £2  7s.  ?  in 
£10  6s.  ? 

6.  How  many  pounds  in  25  shillings?   in  40  shil- 
lings?   in   68  shillings?     How   many  guineas   in   63 
shillings?     How  many  dollars  in  £10. 

7.  What  part  of  a  shilling  is  2  pence ?  3  pence ?  4 
pence  ?  6  pence  ?  18  pence  ? 

8.  What  part  of  a  pound  is  1  shilling  ?  2  shillings  ? 
4  shillings?  2s.  6d.  ?  3s.  5d.  ? 

9.  What  is  the  cost  of  12  articles  at  1  penny  apiece? 
2  pence  apiece  ?  at  4d.  ?  at  3d.  ?  at  lOd.  ? 

10.  What  is  the  cost  of  24  or  2  x  12  articles,  at  2d. 
apiece?  at  3  pence?  4d.  ?  6d.  ?  10d.? 

11.  What  is  the  cost  of  20  articles  at   1  shilling 
apiece?  3s.  apiece?  4s.?  6  shillings?  10  shillings  ? 

12.  What  is  the  cost  of  40  articles,  or  2x20  at  3 
shillings  apiece?  at  4s.  ?  6s.?  10s.? 

13.  What  is  the  cost  of  8  bushels  of  corn  at  2s.  8d. 
a  bushel  ?     What  part  of  a  pound  is  a  penny  ? 


LESSON    VI. 
Measure  of  Time. 

The  natural  units  in  this  table  are  the  day  and  the 
year.  The  smallest  unit,  the  second,  is  the  interval 
between  two  ticks  of  a  seconds  clock,  the  pendulum 
of  the  clock  being  39TV  inches  long. 


130  DENOMINATE  NUMBERS. 

TABLE. 

60  seconds  (sec.)  make  1  minute — marked  min. 
60  minutes  make  1  hour — marked  lir. 
24  hours  make  1  day — marked  dy. 
7  days  make  1  week — marked  wk. 

365  days  make  1  common  year — marked  yr. 

366  days  make  1  leap  year. 

100  years  make  1  century — marked  c. 

The  year  is  divided  into  12  months. 

TABLE. 

1st  month,  January,  has  31  days. 

2d  month,  February,  lias  28  days — in  leap  year,  29. 

3d  month,  March,  has  31  days. 

4th  month,  April,  has  30  days. 

5th  month,  May,  has  31  days. 

6th  month,  June,  has  30  days. 

7th  month,  July,  has  31  days. 

8th  month,  August,  has  31  days. 

9th  month,  September,  has  30  days. 

10th  month,  October,  has  31  days. 

llth  month,  November,  has  30  days. 

12th  month,  December,  has  31  days. 

This  table  is  usually  given  in  the  following  rhyme : 

Thirty  days  hath  September, 
April,  June,  and  November ; 
All  the  rest  have  thirty-one, 
Except  the  second  month  alone, 
And  this  has  twenty-eight,  in  fine, 
Till  leap  year  gives  it  twenty-nine. 

1.  How  many  seconds  in  2  minutes  ?    3  minutes  ? 
4  minutes  ?   10  minutes  ?  in  ^  minute  ?  J  minute  ?  -£% 
of  a  minute  ?  f  of  a  minute  ?  in  T7F  of  a  minute  ? 

2.  How   many  minutes  in   120   seconds?    in   180 


DENOMINATE   NUMBERS.  131 

seconds  ?  in  2  hours  ?  in  3^  hours  ?  in  5  hours  ?   in  If 
hour?  £  an  hour?  £  of  an  hour? 

3.  How  many  hours  in  2  days  ?  3  days  ?  4  days  ?  in 
2£  days?  in  If  day?     How  many  days  in  48  hours? 
72  hours  ?  75  hours  ?  36  hours  ? 

4.  How  many  days  in  4  weeks  ?  in  4  of  a  week  ?  in 
4|-  weeks  ?    in  10  weeks  ?     How  many  weeks  in  28 
days?  56  days?  31  days? 

5.  If  October  commences  on  Monday,  on  what  day 
of  the  week  will  November  begin  ? 

6.  How  many  months  in  2  years  ?    2-j-  years  ?    6 
years  ? 

7.  If   a  man  earns  2|-  cents  a  minute,  how  much 
will  he  earn  in  5  hours  ?     What  part  of  an  hour  is  2 
minutes  ?  3  minutes  ?  15  minutes  ? 

MISCELLANEOUS    TABLE. 

12  units  make  1  dozen. 
12  dozen  make  1  gross. 
20  units  make  1  score. 
24  sheets  of  paper  1  quire. 
20  quires  make  1  ream. 

1.  How  many  units  in  3  dozen  ?  -J-  dozen  ?  5|-  dozen  ? 

2.  How  many  scores  in  40  ?  50  ?  80  ?  60  ?  200  ?   in 
5  dozen  ?  10  dozen  ?  in  4^-  dozen  ? 

3.  What  is  the  cost  of  a  ream  of  paper  at  15  cents  a 
quire?  at  10  cents  ?  12£  cents?  at  5-^  cents  ?  What 
part  of  a  ream  is  half  a  quire  ? 


132  METRIC    SYSTEM. 


CHAPTEK    IX. 

METRIC   SYSTEM    OF   WEIGHTS   AND    MEASURES. 

The  adoption  of  the  French  or  metric  system  of 
weights  and  measures  in  this  country,  has  been  autho- 
rized by  the  Government.  The  great  advantage  of 
this  system  is  the  uniformity  in  the  names  and  in  the 
division  of  the  units  in  the  different  tables.  The  di- 
vision is  the  successive  division  into  tenths,  after  the 
manner  of  the  eagle,  dollar,  dime,  cent,  and  mill  of 
the  United  States  money.  In  addition  to  these  ad- 
vantages, this  system  has  one  table  of  weights  instead 
of  three.  One  table  for  cubic,  liquid,  and  dry  meas- 
ure. The  names  of  the  units  indicate  what  multiple 
or  part  of  the  principal  unit  they  are. 

Long  Measure. 

The  Metre  is  the  principal  unit  or  basis  of  the  table 
of  Long  Measure. 

The  Metre  is  one  yard  3  and  T4T  inches,  or  SOy3^ 
inches. 

TABLE. 

10  millimetres  make  1  centimetre. 

10  centimetres  make  1  decimetre. 

10  decimetres  make  1  metre— 39Ty¥  in. — marked  M. 

10  metres  make  1  decametre- — marked  D  . 

10  decametres  make  1  hectometre — marked  Hm. 

10  hectometres  make  1  kilometre — marked  Km. 

10  kilometres  make  1  myriametre^O-^-  miles,  marked 

Mm. 

The  prefixes,  milli,  centi,  deci,  mean  YoVo  ?  TST >  an^ 
•j^,  respectively.  The  prefixes  deca,  hecto,  kilo, 


METRIC    SYSTEM.  133 

myria,  mean  10,  100,  1000,  10,000,  respectively. 
These  are  used  in  all  the  tables  of  this  system.  In 
commerce  the  expressions  10  metres,  100  metres,  1000 
metres,  10,000  metres,  are  invariably  used  instead  of 
decametre,  hectometre,  kilometre. 

1.  How  many  millimetres  in  2  centimetres  ?  4  cen- 
timetres ?  in  3^  centimetres  ?     What  part  of  a  centi- 
metre is  1  millimetre  ?    2  millimetres  ?     How   many 
centimetres  in  50  millimetres  ?  in  65  millimetres  ? 

2.  What  part  of  a  metre  is  a  millimetre  ?    a  centi- 
metre ?  What  part  of  a  myriametre  is  a  metre  ?  How 
many  metres   make  a  kilometre  ?   how  many  centi- 
metres ?  how  many  decimetres  ? 

3.  Bought  5  decimetres  of  ribbon  at  1  dollar  a  metre, 
what  was  the  cost  ? 

4.  What  is  the  cost  of  30  metres  of  cloth  at  half  a 
dollar  a  decimetre  ?    at  ^  of  a  dollar  a  decimetre  ?    at 
1  dollar  a  decimetre  ? 

5.  A  tower  is  55  metres  high  ;   how  many  centi- 
metres ? 

6.  How  many  miles  in  100  myriametres  ?  40  myria- 
metres  ? 

Square  Measure. 

The  natural  unit  is  the  centiare  or  square  metre, 
equal  to  1-^-  square  yard,  nearly. 

100  centiares  make  1  are  =  100  square  metres. 
10  ares  make  1  decare. 

10  decares  make  1  hectare  =  2-^j-  acres,  2£  acres, 
nearly. 

1.  How  many  square  metres  in  15  centiares?  how 
many  square  metres  in  1  are  ?    in  1  hectare  ?    in    2 
hectares?     How  many  acres  in  100  hectares?     How 
many  square  yards  in  100  square  metres  ? 

2.  What  is  the  cost  of  15  hectares  of  land  at  5  dol- 


134:  METRIC    SYSTEM. 

lars  the  decare?     "What  is  the  cost  of  15  acres  of  land, 
at  50  dollars  the  hectare  ? 

NOTE. — To  make  the  table  of  square  measure  complete,  we  should 
have  the  square  decimetre,  square  centimetre,  and  square  millimetre — 
but  these  are  seldom  used. 

Cubic  Measure. 

The  unit  of  this  table  is  the  cubic  metre  =  1.3  cub. 
yard.  This  is  called  a  stere  when  used  in  measuring 
wood,  or  hectolitre  when  applied  to  liquid  or  dry 
measure,  of  which  the  principal  unit  is  the  litre  or 
cubic  decimetre  =  y9^  of  a  quart,  dry  measure,  and 
1-rtSr  °f  a  quart,  liquid  measure. 

TABLE. 

10  millilitres  make  1  centilitre  =  10  cubic  centi- 
metres. 

10  centilitres  make  1  decilitre. 

10  decilitres  make  1  litre  =  1  cubic  decimetre  =  -fa 
of  a  dry  quart,  D.  M.,  lyf-o  of  a  quart,  L.  M. 

10  litres  make  1  decalitre. 

10  decalitres  make  1  hectolitre. 

10  hectolitres  make  1  kilolitre  or  stere  =  1  cubic 
metre. 

1.  What  part  of  a  kilolitre  is  1  litre  ?  2  litres  ?  1 
decilitre?  1  decalitre?    5  hectolitres?  100  decalitres? 

2.  How  many  millilitres  make  one  decalitre  ? 

3.  How  many  quarts  D.  M.  in  10  litres?     How 
many  quarts,  Liquid  Measure,  in  100  litres  ? 

4.  What  is  the  cost  of  20  hectolitres  of  wine  at  10 
cents  a  litre  ? 

5.  1  hectolitre  is  what  part  of  a  cubic  metre  ?    1 
decilitre  is  what  part  of  a  cubic  decimetre?  is  how 
many  cubic  centimetres  ? 


METBIC    SYSTEM.  135 


Weights. 

The  principal  unit  is  the  gramme,  or  the  weight  of 
a  millilitre  of  pure  water  when  it  is  heaviest. 
The  gramme  is  15^^-  grains  Avoirdupois. 

TABLE. 

10  milligrammes  make  1  ceirtigramme. 

10  centigrammes  make  1  decigramme. 

10  decigrammes  make  1  gramme  =  15-^-  grains 
Avoirdupois. 

10  grammes  make  1  decagramme. 

10  decagrammes  make  1  hectogramme. 

10  hectogrammes  make  one  kilogramme  =  2-^-  Ibs. 
Avoirdupois. 

10  kilogrammes  make  1  myriagramme. 

10  myriagrammes  make  1  quintal. 

10  quintals  make  1  inillier  or  tonneau  =  ly1^-  ton, 
nearly. 

1.  How  many  kilogrammes  in  1  quintal  ?  in  5  quin- 
tals ?  in  4£  quintals  ?     How  many  grammes  in  1  ton- 
neau ?     How  many  myriagrammes  in  3  tonneaux  ? 

2.  1  milligramme  is  what  part  of  two  grammes  ?  of 
3  decagrammes?     1  centigramme  is  what  part  of  a 
kilogramme  ?     How  many  grammes  in   5   kilogram- 
mes ? 

3.  What  is  the  cost  of  two  kilogrammes  of  hay  at 
1  dollar  the  quintal  ? 

4.  At  2  dollars  the  kilogramme,  what  is  the  cost  of 
a  hectogramme  of  tobacco?  of  a  decagramme?  of  15 
grammes  ? 

5.  How   many   pounds    Avoirdupois  in   1   myria- 
gramme  ? 

6.  One  hundred  pounds  how  many  kilogrammes  ? 
One  hundred  kilogrammes  how  many  pounds  ? 


136  TABLE    OF   FRENCH   MONEY. 


7.  What  are  some  of  the  advantages  of  the  metric 
system  of  weights  and  measures  ? 

TABLE  OF  FRENCH  MONEY. 

10  millimes  make  1  centime. 

100  centimes  make  1  franc  =  19  cents  ,,3  mills,  U. 
S.  money. 

20  francs  make  1  louis. 

1.  5  centimes  make  1  sou.     The  franc  is  how  many 
sous  ?     5  francs  are  nearly  one  dollar. 

2.  A  franc  in   silver  weighs  5  grammes.     40  five- 
franc  pieces  in  silver  weigh  one  kilogramme. 

3.  What  is  the  weight  of  1000  silver  francs? 

4.  What  is  the  value  of  a  sum  of  money  weighing 
two  kilogrammes  ? 

5.  What  is  the  weight  of  a  litre  of  pure  water  when 
it  is  heaviest  ?  of  a  decilitre  ?  of  a  decalitre  ? 

6.  Bought  500  kilogrammes  of  hay  at  1  dollar  the 
quintal ;    how  much  did  it  cost  ? 

7.  A  garden  of  500  square  metres  is  sold  for  $50 ; 
how  much  is  that  for  each  are  ? 

8.  A  staircase  of  12  steps  descends  into  a  cellar  2 
metres  and  76  centimetres  deep  j  what  is  the  height  of 
each  step  ? 


PERCENTAGE.  137 


CHAPTER    X. 

PERCENTAGE. 
«  LESSON    I. 

Per  cent,  is  a  contraction  for  per  centum,  which 
means  by  the  hundred,  for  each  hundred,  or  in  a  hun- 
dred. Thus  5  per  cent,  of  any  number  means  5  parts 
in  every  hundred  parts  of  the  number,  or  5  hundredths 
of  ?he  number.  To  take  6  per  cent,  of  a  number  is 
to  take  6  hundredths  of  the  number,  or  6  times  1  per 
cent.,  or  6  times  1  hundredth  of  the  number.  The  5 
and  6  in  5  per  cent,  and  6  per  cent,  are  called  the  rates 
per  cent.  5  per  cent,  and  6  per  cent,  are  merely  other 
forms  for  the  fractions  y^  and  T-|  ¥. 


1.  "What  is  meant  by  per  cent.  \     What  is,  then,  5 
per  cent,  of  any  number  \     What  is  meant  by  the  rate 
per  cent.  ? 

2.  How  much  is  1  per  cent,  of  40  ?  50  ?  60  ?  4  ?  5  ?  6  ? 

3.  How  much  is  1  per  cent,  of  20  ?  2  per  cent,  of  20  ? 

3  per  cent.  ?  5  per  cent.  ?  4  per  cent.  ?  -J  of  1  per  cent.  ? 
J  of  1  per  cent.  ?  2|-  per  cent.  ?  10  per  cent.  ? 

PROCESS.  —  1  per  cent,  of  20  is  y-^-  of  20  =  -gfa  =  \  ; 
•|  0/*  1  jp^/1  ££?^.  i«  ^  of  ^  or  TL-  ;  4  jp^r  cent,  of  20  is 

4  x  -J-,  6>r  |-  ;  10  jt?^  tf^rf.  is  10  x  -J-  =  2. 

4.  How  much  is  10  per  cent,  of  540  ?    320  ?    250  ? 
100  ?     How  much  is  20  per  cent,  of  the  same  num- 
bers? 

5.  How  much  is  100  per  cent,  of  5  ?    of  6  ?    of  any 
number?     How  much  is  200  per  cent,  of  8?   40  ?    of 
any  number? 

6.  Every  per  cent,  fraction  has  what  for  numerator 
and  what  for  denominator  ? 


138  PERCENTAGE. 

ANS. — The  rate  per  cent,  for  the  numerator,  and  one 
hundred  for  the  denominator. 

7.  Which  per  cent,  fractions,  or  wliicli  hundredths, 
can  be  most  readily  reduced  to  simpler  fractions  ? 

ANS. — Those  whose  numerators  are  divisors  of  100. 

8.  What  fraction  of  a  number  having  1  for  its  nu- 
merator is  2  per  cent,  of  the  number  ? 

PROCESS. — 2  per  cent,  of  a  number  is  y-f-g-  of  the  num- 
~ber,  yf  Q-  in  its  simplest  form  =  -^. 

9.  What  is  the  simplest  form  of  4  per  cent.  ?  5  per 
cent.?  10  per  cent.?  20  per  cent.?  25  per  cent.?  50 
per  cent.  ?  75  per  cent.  ? 

10.  What  per  cent,  of  a  number  is  |-  of  the  number  ? 
•|-  is  how  many  hundredths  ? 

11.  What  per  cent,  of  a  number  is  ^  of  it?  -J-  of  it  ? 
TV  of  it?  jV  of  it?  ^  of  it?  TV  of  it?  T^T  of  it?  ^ 
of  it?  £  of  it  ?  yV  of  it  ?  y^  of  it  ? 

12.  Which  fractions  can  be  extended  most  easily  to 
hundredths  ? 

ANS. — Those  of  whose  denominators  100  is  a  mul- 
tiple. 

13.  i  of  a  number  is  what  per  cent,  of  the  number  ? 
Extend  ^  to  hundredths. 

PROCESS. — 100-^-3  is  33^,  hence  %  is  -f- f^ ;  -|-  6/*#  n^ra- 
Z>£r  is  |-|^?  6>r  33|-  jp£r  cent,  of  it. 

11.  What  per  cent,  of  a  number  is  -J-  of  it  ?  |-  of  it? 
TVofit?  TVofit?  f  of  it?  |  of  it?  -Jofit?  f?  -f-  ? 

15.  What  fraction  of  a  number  is  33^-  per  cent,  of 
the  number?  16f  per  cent,  of  it?  12^  per  cent,  of  it? 
8^  per  cent,  of  it  ?  6^  per  cent,  of  it  ?  66f  per  cent.  ? 
87|  per  cent.  ?    60  per  cent.  ?    80  per  cent.  ?    90  per 
cent.  ? 

16.  What  part  of  100  is  16f?  12J?  66f  ?  6J  ?  60? 
80?  37£?  90? 

17.  5  is  what  per  cent,  of  60  ? 

PROCESS. — 5  is  T^-  o/*  60,  <m<$  y1^-  of  a  number  is  T8^- 
of  it,  or  8^-  ^r  cent.  Hence  5  is  8-J-  per  cent,  of  60. 


PERCENTAGE.  139 

The   question,  What  per  cent?  is  simply,  how  many 
hundredth*  f 

IS.  What  per  cent,  of  30  is  5  ?  10?  20?  25?  30? 
15?  2? 

19.  8  is  what  per  cent,  of  16?    32?   36?  40£  ?  60? 
72? 

20.  2  is  what  per  cent,  of  10  ?  8  ?  20  ?  40  ?  50  ?  12  ? 
100?  2?  200? 

21.  %  is  what  per  cent,  of  5  ? 

PROCESS. — ^  is  what  part  of  5  ?  -^  of  5  is  10  per 
cent,  of  it ;  or,  1  is  ^  of  53  or  20  per  cent.,  hence  ^  is 
10  per  cent. 

22.  f  is  what  per  cent,  of  2  ?  f  is  what  per  cent,  of 
10? 

23.  %  is  what  per  cent,  of  f  ? 

PROCESS. — \  is  -f  of  f ;  -J-  #/*  &  number  is  16f 
c'mtf.,  #716?  f  is  5  x  16f ,  <9/'  83-J-  JP^  tftffitf. 

24.  y^  is  what  per  cent,  of  f  ? 

25.  10  is  5  per  cent,  of  what  number  ? 
PROCESS. — 5  per  cent,  is '^  10  is  ^  #/*200, 

cent,  of  200. 

26.  18  is  6  per  cent,  of  what  number? 

PROCESS. — 18  is  Tf¥  of  what  number?  6  in  IS,  3 
times,  3  x  100  —  300;  18  is  6  per  cent,  rt/300. 

27.  20  is  5  per  cent,  of  what  number  ?  4  per  cent.  ? 
10  per  cent.  ?  2-J-  per  cent.  ?  6  per  cent,  of  what  num- 
ber ? 

28.  |-  is  6  per  cent,  of  what  number?  8  per  cent.  ? 
10  per  cent.  ? 

29.  2|-  is  5  per  cent,  of  what  number  ?  12  per  cent.  ? 
6£  per  cent.  ?  33^  per  cent.  ?  40  per  cent.  ? 

30.  What  is  2  per  cent,  of  5  per  cent,  of  a  number  ? 
PROCESS. — 2  per  cent,  of  5  per  cent.  =  -fa  of  5  £>6r 

<?m£.  =  -foper  cent.  =  -^  per  cent. 

31.  How  much  is  50  per  cent,  of  25  per  cent,  of  80? 
PROCESS. — First  take  25  per  cent,  of  80  and  then  50 

per  cent,  of  the  result. 


140  PERCENTAGE. 

32.  25  per  cent,  of  40  per  cent,  is  what  per  cent,  of 
20  per  cent,  of  a  number  ? 

33.  What  fraction  is  5  times  6  per  cent.  ?     12  is  how 
many  times  5  per  cent,  of  40  ?     12  is  6  times  what  per 
cent,  of  50  $ 


cent,  of  50 


LESSON    II. 


1.  What  is  2  per  cent,  of  $10  ?  $8  ?  $20  ?  $50  ?  of 
1  dime  ? 

2.  What   is   5   per  cent,  of   $100?    $300?    $600? 
$1000  ? 

3.  What  is  6  per  cent,  of   $1?    $5?    $50?    $200? 
$150? 

4.  What  is  %  per  cent,  of  $5  ?    $10  ?    100  ?    $500  ? 
$1000  ? 

5.  What  is  4£  per  cent,  of  $100?    $200?    $500? 
$1000  ? 

6.  6  cents  is  what  per  cent,  of  $1  ?  60  cents  is  what 
per  cent,  of  $10  ?     $6  is  what  per  cent,  of  $600  ?    $60 
is  what  per  cent,  of  $1000  ?     $600  is  what  per  cent,  of 
$10,000? 

7.  5  centimetres  is  what  per  cent,  of  a  metre  ?     5 
metres  is  what  per  cent,  of  a  hectometre  ?     1  litre  is 
what  per  cent,  of  a  decalitre  ?     5  decilitres  is  what  per 
cent,  of  a  litre?     A  gramme  is  what  per  cent,  of  a 
kilogramme  ? 

8.  How  much  is  10  per  cent,  of  60  metres  ?  6  per 
cent,  of  20  litres  ?  5  per  cent,  of  40  kilogrammes  ? 

9.  What   is   5   per   cent,   of   £50?    £100?    £600? 
£1000  ? 

10.  How  much  is  4-  per  cent,  of  £1200  ?  4-  per  cent, 
of  £500? 

11.  5  shillings  is  what  per  cent,  of  £1  ?  3d.  is  what 
per  cent,  of  a  shilling? 

12.  What  per  cent,  of  a  yard  is  1  foot  ?  6  inches  ? 

13.  The  number  of  inhabitants  of  a  village  was  last 


PERCENTAGE. 

year  650,  this  year  it  is  30  per  cent,  greater.  What  is 
the  increase  for  the  year  ? 

14.  A   commission  merchant  sells  $600  worth   of 
wheat  jjpr  a  farme^  and  deducts  4  per  cent,  commis- 
sion ;  how  much  does  he  pay  over  to  the  farmer  ? 

15.  I  bought  5  shares  of  railroad  stock,  which  cost 
$100  a  share ;  since  the  purchase  each  share  has  fallen 
10  per  cent. ;  what  are  the  5  shares  worth  now? 

16.  A  house  valued  at  $1200  is  insured  by  a  com- 
pany at  a  cost  of  2  per  cent,  on  this  sum.     How  much 
does  the  owner  of  the  house  pay  the  company  ? 

17.  How  much  is  6£  per  cent,  of  $32  ?  80  bushels 
of  corn  ?  64  barrels  of  flour  ? 

18.  How  much  is  33|-  per  cent,  of  60  gallons  of 
wine  ? 

19.  A  man  owes  $6000  and  is  able  to  pay  only  66f 
per  cent,  of  it.     How  much  do  his  creditors  lose  ? 

20.  Of  a  pound  of  gunpowder  75  per  cent,  is  salt- 
petre, 12|-  per  cent,  is  sulphur,  and  12£  is  charcoal. 
How  many  ounces  of  each  form  the  pound  ? 

21.  Cannon  metal  consists  of  copper  and  tin;  it 
contains  90  per  cent,  of  copper.     "What  per  cent,  of 
tin  ? 

22.  Bought  merchandise  for  $44,  sold  the  same  for 
$22.     What  was  the  loss  per  cent.  ? 

23.  Bought  for  $66,  sold  for  $88.     What  was  the 
gain  per  cent.  ? 

24.  Bought  for  $96,  sold  for  $100.     Gained  what 
per  cent.  ? 

25.  Bought  for  $72,  sold  for  $63.     Lost  what  per 
cent.  ? 

26.  Bought  for  $23,  sold  for  $69.     Gained  what  per 
cent.? 

27.  Bought  for  $64,  sold  for  $56.     Lost  what  per 
cent.  ? 

28.  Bought  for  $48,  sold  for  $44.     Lost  what  per 
cent.  ?    . 


14:2  PERCENTAGE. 

29.  Bought  for  $69,  sold  for  $23.     Lost  what  per 
cent.  ? 

30.  I  bought  merchandise  for  $75  and  sold  it  at  a 
loss  of.  6J  per  cent.     For  how  much  did  I  sell  ?     For 
how  much  would  I  have  to  sell  it  to  gain  6^  per  cent.  ? 

31.  Bought  for  $80,  lost  on  sale  10  per  cent.     Sold 
for  how  much  ? 

32.  Bought  for  $72,  gained  on  sale  33^-  per  cent. 
Sold  for  how  much  ? 

33.  Bought  for  $88,  gained  25  per  cent.     Sold  for 
how  much  2 

34.  Bought  for  $57,  gained  33^  per  cent.     Sold  for 
how  much? 

35.  Bought  for  $57,  lost  33^  per  cent.     Sold  for  how 
much  ? 

36.  Bought  for  $112,  lost  12|-  per  cent.     Sold  for 
how  much? 

37.  A  man  has  200  bushels  of  wheat,  he  sells  75  per 
cent,  of  it  at  $2  a  bushel  and  the  remainder  at  $3  a 
bushel.     How  much  does  he  get  for  the  wheat  ? 

38.  The  rent  of  a  house  was  last  year  $75,  but  this 
year  the  owner  increased  the  rent  20  per  cent.    What 
is  the  rent  this  year? 

39.  Gunpowrder  contains  75  per  cent,  of  saltpetre. 
How  much  saltpetre  in  120  Ibs.  of  it? 

40.  It  requires  16,000  bricks  to  build  a  certain  house. 
If  we  allow  6  J  per  cent,  for  loss  of  bricks  by  breakage, 
how  many  must  be  purchased  for  the  house  ? 

41.  The  rent  of  house  was  increased  this  year  10  per 
cent.,  and  is  now  $99.     What  was  it  before  the  in- 
crease ?     99  is  -^  of  what  number  ? 

42.  A  man  gave  $4000  for  a  house,  and  it  costs  him 
for  repairs  each  year  2  per  cent,  on  the  purchase  money. 
What  is  the  yearly  cost  of  repairs? 

43.  If  30  quarts  of  wine  contain  6  quarts  of  water, 
what  per  cent,  of  water  does  the  wine  contain  ? 

44.  From  $35.50  subtract  20  per  cent. 


PERCENTAGE.  143 

45.  A  merchant  bought  cloth  at  $4-J-  a  yard,  and 
wishes  to  sell  it  at  25  per  cent,  profit,     ^hat  must 
be  the  selling  price  per  yard  ? 

46.  The  gross  weigJtt  of  an  article  of  merchandise 
is  its  weight  including  the  box,  barrel,  or  bag  which 
contains  it.     The  net  weight  is  the  weight  after  the 
weight  of  the  box,  &c.,  is  deducted.     The  allowance 
for  the  weight  of  the  box  or  barrel,  &c.,  is  called  the 
tare. 

47.  The  gross  weight  of  some  merchandise  is  160 
Ibs.,  and  the  tare  is  6£  per  cent.     "What  is  the  net 
weight  ? 

48.  Gross  weight  210   pounds,  tare  10  per   cent. 
What  is  the  net  weight  ? 

49.  Gross  weight  250  pounds,  net  weight  230  pounds. 
What  per  cent,  of  tare? 

50.  Gross   weight   220   pounds,   tare   5    per   cent. 
What  is  the  net  weight  ? 

51.  A  man  bought  a  horse  for  $100,  and  sold  him 
for  $300.     What  per  cent,  did  he  gain  ? 

52.  The  person  who  bought  the  horse  at  $300  sold 
it  for  $100.     What  per  cent  did  he  lose  ? 

53.  $5  is  what  per  cent,  of  $6  ?     $6   is  what  per 
cent,  of  $5. 

54.  Bought  an  article  for  2s.  6d.,  and  sold  it  for  3s. 
6d.     What  was  the  gain  per  cent.  ? 

55.  Bought  an  article  at  $4.30,  sold  it  for  $5.30. 
What  was  the  gain  per  cent.  ? 

56.  Bought  5  pounds*of  sugar  at  Is.  6d.,  and  sold  it 
at  5d.  a  pound.     What  per  cent,  did  I  gain  ? 

57.  Sold  a  horse  for  $03,  and  lost  12J  per  cent,  of 
what  he  cost  me.     What  did  he  cost  ? 

SOLUTION. — Lost  -J-.     Hence  $63  is^of  his  cost. 

58.  If  $150  in   currency  are  worth  only  $100  in 
gold,  what  percentage  is  lost  in  receiving  in  currency 
a  debt  payable  in  gold?     $50  is  what  part  of  $150 '} 
%  of  a  number  is  what  per  cent,  of  it  ? 


144  PERCENTAGE. 

59.  If  $100  in  gold  are  worth  $150  in  currency, 
at  what  per  cent,  premium  is  gold  ? 

NOTE. — Let  the  pupil  frame  questions  similar  to  the  above,  and 
give  the  ablutions. 

LESSON    III. 
SIMPLE    INTEREST. 

Interest  on  money  is  the  sum  which  the  borrower 
pays  the  lender  for  the  use  of  the  borrowed  money. 
The  money  loaned  is  called  the  Principal.  The  sum 
of  the  principal  and  interest  is  called  the  Amount. 

Simple  interest  is  computed  at  so  much  for  a 
given  time — usually  a  year — for  every  hundred  dollars, 
pounds,  or  francs  of  the  loan /  in  other  words,  at  a 
given  rate  per  cent,  per  year  on  the  principal. 

In  computing  for  shorter  periods  than  a  year,  thirty 
days  are  usually  considered  a  month,  and  twelve 
months  a  year. 

1.  At  6  per  cent.,  what  is  the  interest  on  $1  for  a 
year?  $5  for  a  year?  $10?  $50?  $500?  $20? 

2.  At  7  per  cent.,  what  is  the  interest  on  $1  for  one 
year  ?  on  $2  for  one  year  ?  on  $5  ?  $10  ?  $20  ?  $100  ? 
$1000?  $300? 

3.  At  8  per  cent.,  what  is  the  interest  on  $1  for  one 
year  ?  on  $10  for  one  year  ?  on  $100  ?  on  $1000  ?  on 
$5?  $50?  $500?  $5000? 

4.  At  10  per  cent.,  what  id  the  interest  on  $1  for 
one  year  ?    on  $10  ?    on  $5  ?    $100  ?   $1000  ?    $500  ? 
$3000  ? 

5.  At  3£  per  cent.,  what  is  the  interest  on  $1  for 
one  year?   on  $10?    on  $100?    on  $1000?    on  $60 1 
on  $600  ? 

6.  What  is  the  interest  on  $1  for  2  years  at  6  per 
cent.  ?  7  per  cent.  ?  8  per  cent.  ?  10  per  cent.  ?  3|-  per 
cent.  ?  4J  per  cent.  ? 


SIMPLE   INTEREST.  14:5 

7.  What  is  the  interest  on  $20  for  3  years  at  5  per 
cent.  ?  6  per  cent.  ?  7  per  cent.  ?  8  per  cent.  ?  10  per 
cent.  ?  3^  per  cent.  ? 

8.  What  is  the  interest  on  $200  for  3|-  years  at  5 
per  cent.  ?  7  per  cent.  ?  10  per  cent.  ?  3J  per  cent.  ?  6 
per  cent.  ? 

9.  What  is  the  interest  on  £25   for  one  year  at  6 
per  cent.  ? 

10.  What  is  the  interest  on  £20  for  10  years  at  5  per 
cent.  ? 

11.  What  is  the  interest  on  $1  for  1  month  at  6  per 
cent.  ? 

ANS. — rV  of  6  cents,  or  -J-  cent  =  5  mills. 

12.  TV  of  6  per  cent,  of  any  number  is  what  per  cent, 
of  that  number?  6  per  cent,  per  year  on  a  sum  of 
money  is  what  per  cent,  per  month  on  it  ? 

13.  What  is  the  interest  on  $10  for  1  month  at  6  per 
cent.?  on  $100?  on  $1000?  on  $3 ?  $30?  $25? 

14.  What  is  the  interest  on  $1  at  6  per  cent,  per 
year  for  2  months?  3  months?  5  months?  9  months? 
7  months  ?  On  $10  for  2,V  months  ?  on  $50  for  3  months  ? 
on  $100  for  6  months?  *8  months  ?  10  months? 

15.  What  is  the  interest  on  $1  for  1  month  at  8  per 
cent.  ?  7  per  cent.  ?  10  per  cent.  ?  5  per  cent.  ?     8  per 
cent,  for  12  months  is  what  per  cent,  for  1  month  ? 

16.  How  much  is  f  of  1  per  cent,  of  $1  ?  T7j  of  1 
per  cent,  of  $1?  |-  of  1  per  cent,  of  $1  ?  T5T  of  1  per 
cent,  of  $1  ? 

17.  What  is  the  interest  on  $100  for  7  months  at  6 
per  cent.  ?  7  per  cent.  ?  10  per  cent.  ?  8  per  cent.  ? 

18.  What  is  the  interest  on  $84:  at  7  per  cent,  for  2 
months  ?  3  months  ?  6  months  ?  8  months  ?  10  months  ? 
9  months?     2,  3,  6,  8,  9,  and   10  form  what  parts,  re- 
spectively, of  12  ? 

19.  At  6  per  cent.,  what  is  the  interest  on  $50  for  4 
years  and  3  months  ?  for  3  years  and  2  months  ?  for  3 
years  and  6  months  ?    2  years  and  4  months  ?  1  year 
and  7  months?  0  years  and  9  months? 


14:0  SIMPLE    INTEREST. 

20.  What  is  the  interest  on  $200  for  3  years  and  5 
months  at  7  per  cent.  ?    8  per  cent.  ?  10  per  cent.  ?  3£ 
per  cent.  ?  5  per  cent.  ? 

21.  The  interest  on  any  sum  for  a  month  is  what 
part  of  the  interest  for  one  year  ?     The  interest  on  any 
sum  for  1  day  is  what  part  of  the  interest  on  it  for  1 
month  ?  on  any  sum  for  2  days  ?  3  days  ?  5  days  ?  6 
days  ?  15  days  ?  10  days  ?  9  days  ?  12  days  ?  18  days  ? 
24  days  ? 

22.  *  At  6  per  cent.,  what  is  the  interest  on  $1  for  6 
days  ?  on  $5  for  6  days  ?  on  $50  for  6  days  ?  on  $500 
for  6  days  ?   on  $5000  for  6  days  ?   on  $1  for  12  days  ? 
24  days  ?  60  days  ?  3  days  ?  5  days  ?  7  days  ? 

23.  AVhat  is  the  interest  on  $100  for  4  months  and 
15  days  at  7  per  cent.  ?  8  per  cent.  ?  6  per  cent.  ?  5 
per  cent.  ?  10  per  cent.  ?  3|-  per  cent.  ? 

24.  What  is  the  interest  on  $50  at  8  per  cent,  for  3 
days?  5  days?  10  days?  12  days?  24  days?  8  days? 

25.  At  6  per  cent.,  what  is  the  interest  on  $1  for  60 
days?   on  $10  for  60  days?   on  $50?  on  $250?   on 
$144? 

26.  At  6  per  cent.,  what  is  the  interest  on  $120  for 
3  years  2  months  and  12  days  ?  for  4  years  5  months 
and  10  days? 

27.  What  is  the  interest   on  $300   for  2  years  4 
months  and  15  days  at  7  per  cent.  ?  8  per  cent.  ?  10 
per  cent.  ?  5  per  cent.  ? 

28.  The  interest  on  a  certain  sum  for  a  given  time 
at  6  per  cent,  is  $18.     What  is  it  at  5  per  cent.  ?  at  1 
per  cent.  ?     What  part  of  6  per  cent,  is  5  per  cent.  ? 

29.  The  interest  on  any  principal  at  1  per  cent,  for 
one  year  is  what  part  of  the  principal  ?  at  4  per  cent., 
is  what  part  of  the  principal  ?  at  12  J  per  cent.  ? 


SIMPLE   ESTEREST. 


LESSON    IV. 

1.  What  is  the  amount  of  $1  at  6  per  cent,  for  1 
year  ?     What  is  the  amount  of  $10  at  5  per  cent,  for 
1  year  ? 

ANS.— 1.  $1.06;  2.  $10.50. 

2.  What  is  the  interest  on  $150  for  2  years  at  6  per 
cent.  ?     What  is  the  Amount  ? 

3.  At  6  per  cent,  what  is  the  amount  of  $200  for 
60  days  ?     What  is  the  amount  of  $80  for  90  days  ? 

4.  What  is  the  amount  of  $50  for  2  years  and  6 
months  at  4  per  cent.  ?  7  per  cent.  ?  8  per  cent.  ?  10 
per  cent.  ?  5  per  cent.  ? 

5.  What  is  the  amount  of  $200  for  2  years  6  months 
and  15  days  at  6  per  cent.  ?  at  10  per  cent.  ?  5  per 
cent.  ? 

6.  What  is  the  amount  of  $300  for  9  years  at  8  per 
cent.  ?  at  7  per  cent.  ? 

7.  At  what  rate  per  cent,  will  $100  yield  $5  interest 
in  1  year  ?     At  what  rate  per  cent,  will  $20  yield  $1 
in  one  year?     $5  is  what  per  cent,  of   $100  ?     $1  is 
what  per  cent,  of  $20  ? 

8.  The  interest  on  $200  for  2  years  is  $14.     What 
is  the  rate  per  cent.  ? 

SOLUTION. —  What  is  the  interest  for  1  year  f     $7  is 
what  per  cent,  of  $200  ? 

9.  If  the  interest  on  $60  for  1  year  and  4  months  is 
$4,  what  is  the  rate  per  cent.  ? 

10.  If  the  interest  on  $90  for  3  years  and  4  months 
is  $30,  what  is  the  rate  per  cent.  ? 

11.  At  what  per  cent,  will  $20  gain  $6  in  1  year  and 
3  months  ? 

12.  At  what  per  cent,  will  $100  gain  $100  interest, 
or  double  itself,  in  20  years  ?  in  10  years  ?  in  5  years  ? 

16  J  years  ?  in  12-|-  years  ?  in  14f  years  ?     At  what 


'" 


148  SIMPLE    INTEREST. 

rate  per  cent,  will  any  sum  double  itself  in  the  times 
mentioned  ? 

SOLUTION. — Tjf  $100  gain  $100  in  20  years,  how  much 
must  it  gain  in  one  year?  If  $100  gain  $100  in  20 
years,  liow  many  dollars  interest  will  $500  gain  ? 

13.  If  the  interest  on  $60  at  5  per  cent,  is  $6,  how 
long  has  it  been  on  interest  ? 

SOLUTION. — The  interest  on  $60  for  one  year  at  5  per 
cent,  is  how  much  f  $3  is  contained  in  $6  how  many 
times  f 

14.  In  what  time,  at  6  per  cent.,  will  $10  gain  $6 
interest  ?      In  what  time,  at  8  per  cent.,  will  $50  gain 
$12  interest? 

15.  In  what  time,  at  7  per  cent.,  will  $40  gain  $6 
interest  ? 

16.  In  what  time,  at  10  per  cent.,  will  $50  gain 
$1241 

17.  In  what  time  will  $1  gain  $1  interest,  at  6  per 
cent.  ? 

SOLUTION. — At  1  per  cent.  $1  will  double  itself,  or 
make  WO  per  cent.,  in  100  years.  Hence,  at  6  per  cent., 
it  will  double  itself  in  -i-g-0-  years,  or  1 6f  years. 

18.  In  what   time  will  a  given  principal  double 
itself  at  5  per  cent.  ?  7  per  cent.  ?  8  per  cent.  ?  10  per 
cent.  ?  3|  per  cent.  ?  25  per  cent.  ? 

19.  In  what  time  will  $100  gain  $200  interest,  at  6 
per  cent.  ?   at  7  per  cent.  ?    at  8  per  cent.  ?    at  10  per 
cent.  ? 

20.  What  principal,  at  6  per  cent.,  will,  in  1  year, 
gain  $48  interest  ?     48  is  yf^  of  what  number  ? 

21.  What  principal,  at  6  per  cent.,  will,  in  4  years, 
yield  $48  interest  ?     $48  in  4  years  is  how  much  in 
1  year  ?     12  is  y^  of  what  number  ? 

22.  What  principal,  in  1  year  and  4  months,  at  8 
per  cent.,  will  yield  $32  interest  ? 

23.  What  principal,  in  2  years  and  3  months,  at  7 
per  cent.,  will  yield  $21  interest? 


SIMPLE    INTEREST.  149 

24.  What  principal,  in  7  months  and  15  days,  at  8 
per  cent.,  will  yield  $40  interest  ? 

25.  What  principal,  in  7  months  and  15  days,  at  4 
per  cent.,  will  yield  $40  interest  ? 

26.  What  principal,  at  10  per  cent.,  will  in  3  years 
and  6  months  yield  $60  interest  ? 

27.  What  principal,  at  10  per  cent.,  will  amount  to 
$55  in  one  year  ? 

SOLUTION. — The  interest  is  ^-  of^  file  principal. 
The  amount  is  therefore  ^  of  the  principal,  and  $55 
is  ±1.  of  $50. 

28.  What  principal  will,  in  12  years,  at  5  per  cent., 
amount  to  $120  ? 

29.  What  principal  will,  in  5  years,  at  7  per  cent., 
amount  to  $85  ? 

30.  What  principal,  at  8  per  cent.,  will  amount  to 
$116  in  4  years? 

31.  What  principal,  at  10  per  cent.,  will  amount  to 
$44  in  2  years  ? 

32.  What  principal,  at  10  per  cent.,  will  amount  to 
$200  in  10  years  ?     What  at  5  per  cent,  will  amount 
to  $200  in  20  years  ?     What  at  6  per  cent,  will  amount 
to  $200  in  16|  years  ?     What  at  8  per  cent,  to  $200  in 


33.  The  present  worth  of  a  debt  due  at  some  future 
time  without  interest  is  equal  to  a  principal  which  put 
at  interest  will  amount  to  the  debt  when  it  is  due. 

The  discount  on  a  debt  is  the  difference  between  the 
debt  and  its  present  worth. 

34.  What  is  the  present  worth  and  the  discount  of 
$45    due   live   years   hence  without  interest,  money 
'being  at  10  per  cent.  ? 

SOLUTION. —  What  principal  at  W  per  cent,  will  in 
5  years  amount  to  $45  ?  What  is  the  difference  be- 
tween $45  and  $30  ? 

35.  What  is  the  present  worth  of  $100  due  in  90 
clays,  money  being  at  6  per  cent.  ?  at  7  per  cent.  ?  at 


150  SIMPLE   INTEREST. 

8  per  cent.  ?  at  10  per  cent.  ?  at  5  per  cent.  ?     What 
is  the  discount  in  each  case  ? 

36.  What  is  the  discount  of  $100  due  in  10  years, 
money  being  at  10  per  cent.  ?  due  in  12  years  at  8  per 
cent.  ?  in  14|  years  at  7  per  cent.  ? 

37.  What  is  the  discount  of  $600  dollars  due  in  5 
years  at  4  per  cent.  ?  due  in  4  years  at  5  per  cent.  ? 
due  in  2  years  at  10  pe^  cent.  ?  due  in  1  year  at  20 
per  cent.  ?  10  years  at  2  per  cent.  ? 

38.  What  is  the  discount  on  $160,  due  in  4  months, 
at  6J  per  cent.  ? 

39.  What  is  the  discount  on  $144,  due  in  15  days,  at 
8  per  cent.  ? 

40.  What  sum  placed  at  interest  will  amount  to 
$960  in  6  years  at  10  per  cent.  ?  in  12  years  at  5  per 
cent.  ?  in  10  years  at  6  per  cent.  ? 

41.  When  banks  lend  money,  they  usually 'discount 
or  take  off  from  the  sum  for  which  the  borrower  gives 
his  note,  the  interest  for  the  time,  and  lend  the  bor- 
rower the  difference. 

42.  What  is  the  bank  discount  on  $100  for  6  months, 
at  6  per  cent. 

SOLUTION. — The  bank  discount  is  the  interest  on  $100 
for  6  months  at  §  per  cent.,  $3.  The  'borrower  gives 
his  note  for  $100  without  interest,  and  gets  $97. 

43.  What  is  the  bank  discount   on  $5,000  for  60 
days,  at  6  per  cent.  ?  at  10  per  cent  ? 

44.  I  wish  to  borrow  $990  from  a  bank  for  60  days, 
at  6  per  cent.     For  what  sum  must  I  give  my  note 
to  the  bank  ? 

45.  What  is  the  bank  discount  on  $100  for  90  days, 
at  6   per  cent.  ?   7  per  cent.  ?  8   per  cent.  ?    10   pel1 
cent.  3 


MISCELLANEOUS.  151 


CHAPTEE    XI. 

MISCELLANEOUS. 
L  E  S  S  0 1ST   I  .—Proportional  Parts. 

1.  Divide  30  into  two  numbers  proportional  to  the 
numbers  1  and  2  ;  or  parts  which  shall  be  to  each 
other  as  2  is  to  1 ;  or  which  divided  one  by  the  other 
shall  give  the  same  quotient  as  2  -f-  1. 

SOLUTION. — 2  and  1  make  3.  Divide  30  into  3  equal 
parts,  and  2  of  tJiese  parts  will  make  one  of  the  num- 
berSj  and  the  remaining  part  will  be  the  other.  We 
thus  get  20  and  10,  and  20  -^  10  =  2-f-l. 

2.  Divide  35  into  two  parts  proportional  to  3  and  4. 

3.  Divide  50  into  two  parts  proportional  to  4  and  5. 

4.  Divide  40  into  two  such  parts  that  the  first  shall 
be  4  times  the  second. 

5.  I  divided  60  oranges  between  Charles  and  Frank, 
giving  Charles  7  for  each  5  I  gave  to  Frank ;   how 
manj  did  each  receive  ? 

6.  I  wish  to  divide  $190  between  two  men,  giving 
$10  to  one  for  each  $9  which  I  give  to  the  other ;  how 
much  will  each  receive  ? 

7.  Divide  36  into  3  parts  wliich  shall  be  to  each 
other  as  the  numbers  2,  3,  and  4. 

8.  Three  men  worked  together  for  a  certain  time  at 
diiferent  wages.     The  first  at  $3  a  day,  the  second  at 
$2  a  day,  and  the  third  at  $1  a  day.     At  the  end  of 
the  time  they  received  $60  together.     What  is  the 
share  of  each  ? 

9.  Divide  80  into  two  parts  which  shall  be  to  each 
other  as  ^  is  to  |-. 


152  MISCELLANEOUS. 

SOLUTION. — J-  is  to  %  means  -J-  -~-  -J-,  or  3  -4-  2. 
divide  80  -info  2  jpa/'te  proportional  to  3  a?irf  2  ; 
£  is  to  -J-  a-9  f  is  to  f — as  3  &s  to  2. 

10.  Divide  34  into   two    parts  proportional   to  •£ 
and  f . 

11.  Divide  51  into  three  parts  proportional  to  f ,  £, 
and  J. 

NOTE — Fractions  reduced  to  a  common  denominator  are  to  each  other 
as  their  numerators. 

12.  In   an  orchard  containing  96  trees  there  are  5 
apple  trees  for  every  4  peach  trees,  and  4  peach  trees 
for  every  3  pear  trees  ;  how  many  trees  of  each  kind 
in  the  orchard  ? 

„  13.  Divide  $200  among  A,  B,  and  C,  so  that  B 
may  get  twice  as  much  as  A,  and  C  three  times  as 
much  as  A. 

14.  Two  men  cut  a  certain  amount  of  wood  for  $55  ; 
one  cuts  6  cords  while  the  other  cuts  5  ;  what  should 
be  the  share  of  each  one  ? 

15.  Two  men  engage  in  trade  together  ;  one  puts  in 
$8,000  for  a  year  and  the  other  $8,000  for  a  year.     At 
the  end  of  the  year  the  profits  are  $2,200 ;  how  much 
should  each  receive  ? 

16.  Three  men  engage  in  trade ;  A  puts  in  $2,000 
for  one   year,  B  $3,000  for   the  same  time,  and   C 
$5,000  for  the  same  time  also.     They  gain  together 
$1,200  ;  what  is  the  share  of  each  in  the  profits  ? 

17.  Three  men  hire  a  pasture  for  a  given  time  for 
$60.     One  of  them  keeps  3  cows  in  it  for  the  whole 
time,  the  second  keeps  5  cows,  and  the  third  keeps  4 
cows.     "What  part  of  the  $60  should  each  one  pay  ? 

REMARK. — When  partners  put  sums  of  money  in 
business  together  for  different  times,  their  respective 
shares  of  tie  profit  or  loss  are  sometimes  by  agree- 
ment proportional  to  the  amounts  put  in  multiplied  by 
the  times  respectively. 

18.  Two  men  form  a  partnership ;  A  puts  in  $200 


MISCELLANEOUS.  153 

for  5  months,  and  B  puts  in  $300  for  2  months.     They 
gain  $96  ;  what  is  each  man's  share  of  the  profits  ? 

19.  In   trading    together,  A    puts   in    $100    for   6 
months,  and  B  $50  for  4  months.     They  lose  $64 ; 
what  is  each  one's  share  of  the  loss? 

20.  Three  men  rent  a  pasture  for   $60.     In  it  A 
pastures  5  cows  for  2  weeks,  B  pastures  two  cows  for 
4  weeks,  and  C  4  cows  for  3  weeks ;  how  much  must 
each  one  pay  for  the  pasturage? 

21.  Eeduce  the  ratio  of  7,000  ~  5,000  to  its  simplest 
form. 

22.  Reduce  the  ratio  of *jj-  -f-  f  to  the  ratio  of  two 
whole  numbers. 

23.  Change  the  terms  of  the  ratios  25  -f-  5,  40  -f-  8, 
120  -f-  12,  4  —  3,  11  -r-  5,  in  such  manner  that  each 
divisor   becomes  1  and    each   quotient   remains   un- 
changed. 

24.  What  number  is  that  to  which  if  its  half  and 
fourth  be  added  the  sum  will  be  28  ? 

25.  A  and  B  are  50  miles  apart,  traveling  toward 
each  other.     When  they  meet  it  is  found  that  £  of  the 
distance  traveled  by  A  is  -|  of  that  traveled  by  B  ; 
what  distance  had  each  traveled  ? 

26.  ^  of  the  time  past  noon  =  \  of  the  time  before 
midnight ;  what  o'clock  is  it  ? 

SOLUTION. — Divide  12  into  two  parts,  which  bear 
to  each  other  the  ratio  of  \  to  -J,  i.  <?.,  3  to  5. 

27.  What  o'clock  is  it  when  the  time  past  noon  is 
-|  of  the  time  before  midnight  ? 

28.  What  number  added  to  twice  itself  gives  24  ? 

LESSON    II. 

1.  The  sum  of  two  numbers  is  25,  and  the  greater 
exceeds  the  less  by  11  what  are  the  fwo  numbers? 

SOLUTION. — Twice  the  less  number  added  to  11  is 
equal  to  25.  Hence,  twice  the  less  number  is  equal  to 


154  MISCELLANEOUS. 

the  difference  'between  25  and  11,  or  the  less  number 
==  |-  £/us  difference  =  7.  -7%£  greater  is  then  7+11 
=  18.  ^ 

2.  Divide  40  into  two  such  parts  that  the  greater 
shall  exceed  the  less  by  22. 

3.  Of  a  pole  50  feet  long,  -J-  as  much  is  in  the  mud 
as  in  the  water,  and  3  times  as  much  in  the  air  as  in 
the  mud  and  water ;  how  much  in  the  mud,  air,  and 
water,  respectively  ? 

4.  A  boy  being  asked  how  many  apples  he  had,  re- 
plied, if  besides  what  he  then  had,  he  also  had  {-  and  -J- 
as  many  more  he  would  hafre  57.     How  many  had  he? 

5.  One  man  can  do  a  piece  of  work  in  5  days,  and 
another  in  20  days ;  in  what  time  can  both  do  the 
work  ? 

SOLUTION. — The  first  does  -J-  of  the  work  in  one  day, 
and  the  second  •£$.  Together  they  do  ^  -f  -fa  =  -^  =  J 
of  the  work  in  one  day  •  or  the  whole  work  in  4  days. 

6.  A  can  reap  a  field  of  wheat  in  10  days,  B  can 
reap  the  same  field  in  90  days ;  in  what  time  can  they 
both  do  the  work  together  ? 

7.  A  can  do  a  piece  of  work  in  4  days,  B  can  do  it 
in  6  days,  and  C  in  3  days ;  in  what  time  can  they  all 
do  it  together  ? 

8.  A  barrel  of  flour  lasts  a  certain  man  and  his  wife 
12  weeks,  but  when  the  man  is  absent  it  lasts  his  w.ife 
20  weeks ;  how  long  would  a  barrel  last  the  man  alone  ? 

9.  A  and  B  can  do  a  piece  of  work  in  12  days,  but 
when  B  is  absent  A  requires  20  days  to  do  the  same 
work ;  how  long  would  it  require  B  to  do  the  work 
alone  ? 

10.  A  and  B  can  do  a  piece  of  work  in  3  days,  A 
and  C  in  4  days,  B  and  C  in  5  days ;  in  what  time 
can  each  one,  working  alone,  do  it?     In  what  time  can 
they  all  do  the  work  together  ? 

11.  A  cistern  is  discharged  by  three  pipes ;  the  first 
and  second  together  will  discharge  it  in  one  hour,  the 


MISCELLANEOUS.  155 

first  and  third  together  in  2  hours,  the  second  and  third 
in  3  hours;  in  what  time  can  they  all  three  discharge 
it  ?  In  what  time  can  each  discharge  it  ? 

12.  A,  B,  and  C  can  perform  a  certain  work  in  4 
days,  C  working  alone  can  do  the  same  work  in  12 
days,  and  B  alone  in  10  days ;  in  what  time  can  C  do 
the  work  ?     In  what  time  can  B  and  C  together  ? 

13.  One  pipe  leading  into  a  cistern  can  fill  it  in  3 
hours,  and  another  pipe  leading  from  a  cistern  can 
empty  it  in  5  hours.     If  the  water  is  running  through 
both  pipes  at  the  same  time,  in  what  time  will  the  cis- 
tern be  filled  ? 

14.  A  man  and  his  wife  would  eat  one  cwt.  of  flour 
in  18  days ;  after   living   together  for  10  days,  the 
woman  could   eat  the  remainder  in  30  days ;  in  how 
many  days  does  the  man  eat  one  cwt.  of  flour  ? 

15.  A  man  wishes  to.  give  10  cents  apiece  to  some 
beggars,  but   finds  he  has  not  money  enough  by  9 
cents :  he  gives  them,  however,  8  cents  apiece,  and* 
has  5  cents  remaining;  how  many  beggars  were  there  ? 

16.  A  can  reap  f  of  a  certain  field  in  2  days,  and 
Bean  reap  T9Q- of  the  same  field  in  3  days;  in  what 
time  can  they  both  together  reap  the  whole  field? 

17.  A  can  reap   1  acre  in  2  hours,  B  can  reap  2 
acres  in  3  hours ;   in  what   time  will  they  together 
reap  14  acres  ?    And  if  the  payment  is  half  a  dollar  an 
acre,  how  much  will  each  one  receive  ? 

18.  A  man  hired  a  laborer,  giving  him  one  dollar 
a  day  for  each  day  that  he  labored,  and  exacting  25 
cents  for  each  day  that  he  was  idle.     At  the  end  of 
18  days  he  received  8  dollars ;  how  many  days  did 
he  labor  and  how  many  was  he  idle  ? 

19.  The  sum  of  two  numbers  is  20,  and  three  times 
the  less  is  less  by  15  than  twice  the  greater  part ;  what 
are  the  two  parts  ? 

20.  The  sum  of  three  numbers  is  20.     The  sum  of 
the  first  and  second  is  equal  to  the  third,  and  half  the 


156  MISCELLANEOUS. 

sum  of  the   first   and   third   is   equal   to   twice  the 
second. 

21.  If  f  of  a  number  is  increased  by  10  the  sum  is 
40.     What  is  the  number  ? 

22.  If  |-  of  a  number  be  diminished  10  the  remainder 
is  25.     What  is  the  number? 

23.  A  can  do  a  piece  of  work  in  -J-  of  a  day  and  B 
in  J  of  a  day.     How  long  will  it  take  both  to  do  it  ? 

24.  Three  men,  A,  B,  and  C,  can  do  a  piece  of 
work  in  6  days  ;  A  can  do  it  alone  in  10  days,  and  B 
in  12  days ;  in  how  many  days  can  C  do  the  work  ? 

25.  A  cistern  of  200  gallons  is  emptied  in  30  minutes 
by  3  pipes.     The  first  pipe  discharges  -J-  gallon  in  a 
minute,  the  second  l£  gallon  ;  how  much  does  the 
third  pipe  discharge  per  minute  ? 

26.  A  can  do  a  piece  of  work  in  2  days,  B  in  3 
days,  C  in  4  days,  and  D  in  6  days.     In  what  time 
can  they  all  together  do  the  wrork  ? 

LESSON    III. 

1.  Eggs  are  sold  at  the  rate  of  5  for  3  pence.     At 
what  rate  must  they  have  been  bought  in  order  to 
make  this  a  gain  of  20  per  cent.  ? 

2.  A  man  mixed  20  gallons  of  ale  worth  11  pence  a 
gallon  with  30  gallons  of  beer  worth  8  pence  a  gallon  ; 
what  is  the  price  per  gallon  of  the  mixture  ?     What 
must  be  the  selling  price  per  gallon  in  order  to  gain 
20  per  cent.  ? 

3.  A  merchant  sold  his  flour  at  various  prices — one 
barrel  for  $9,  a  second  for  $10,  a  third  barrel  for  $12, 
a  fourth  barrel  for  $11,  a  fifth  barrel  for  $8.     What 
was  the  average  price  per  barrel  ? 

4.  What  is  the  average  of  the  numbers  7,  8,  12,  7J 
and  5iJ-? 

5.  If  a  pole  30  feet  high  at  a  given  time  of  day  casts 
a  shadow  15  feet  long,  how  long  a  shadow  will  be  cast 


MISCELLANEOUS.  157 

at  the  same  time  by  a  pole  13  feet  high  ?  How  high 
a  pole  will  at  the  same  time  cast  a  shadow  20  feet 
long  ? 

6.  If  ten  oxen  eat  2  tons  of  hay  in  30  days,  how 
many  oxen  will  it  take  to  eat  16  tons  in  50  days  ?  in 
60  days  ?  in  75  days  ? 

7.  If  6  men  can  dig  a  ditch  of  a  certain  depth  and 
width  8  rods  long  in  4  days,  how  long  will  it  take  12 
men  to  dig  a  ditch  of  the  same  depth  and  width,  40 
rods  long  ?  72  rods  long  ? 

8.  The  posts  of  an  electric  telegraph  by  the  side  of 
a  railway  are  placed  at  intervals  of  60  yards  ;  at  what 
rate  must   a  train  move  to  pass  10  of  these  in  20 
seconds  ? 

9.  The  work  of  54  men  in  8  days  is  equal  to  that 
of  16  workmen  in  how  many  days '{ 

10.  If  15  workmen  in  30  days  finish  100  rods  of  a 
wall  of  certain  thickness  and  height,  how  many  rods 
can  6  workmen  finish  in  81  days  ?     How   many  rods 
60  workmen  in  9  days  ?  9  workmen  in  90  days  ?  12 
workmen  in  54  days? 

11.  If  12  workmen  can  dig  48  rods  of  a  ditch  of 
certain  width  and  depth  in  6  days,  in  how  many  days 
can  50  workmen  dig  600 rods  ?  30  workmen  180  rods? 

12.  A  small  steam-engine  of  one  horse-power  burns 
in  an  hour  20  pounds  of  coal ;  how  many  hundred- 
weight will  it  require  in  5  days? 

13.  When  $100  in  gold  is  worth  $148  in  currency, 
how  much  is  $5  in  gold  worth? 

14.  What  is  the  price  of  240  articles  at  3  pence 
apiece  ?    at  8  pence  apiece  ?      What  is  the  price  of 
300  articles  at  5  shillings  each  ?  of  240  articles  at  3 
cents  each  ?  of  300  articles  at  $5  each  ? 

15.  A  fox  is  50  rods  ahead  of  a  hound,  and  is  run- 
ning 70  rods  a  minute ;  the  hound  is  pursuing  at  the 
rate  of  75  rods  a  minute ;  in  how  many  minutes  will 
the  hound  overtake  the  fox  ? 


158  MISCELLANEOUS. 

10.  If  an  army  marches  130  miles  in  6|-  days, 
marching  9  hours  a  day,  how  far  can  it  march  in  4£ 
days  by  marching  10  hours  a  day? 

LESSON    IV. 

1.  5  is  ^-.     Hence  to  multiply  a  number  by  5",  an- 
nex 0  and  divide  by  2.     42  x  5  =  420  -=-  2  =*  210. 

Multiply  94,  48,  63,  75,  and  448  by  5. 

2.  25  =  J-jp-.     Hence  to  multiply  a  number  by  25, 
annex  00  and  divide  by  4.     36  x  25  =  3600  -^-  4  —  900. 

Multiply  36,  72,  68,  84,  and  79  by  25. 

3.  125  =  —^-.     Hence  to  multiply  a  number  by 
125  we  annex  000  and  divide  by  8. 

4.  Multiply  56,  72,  88,  and  91  by  125. 

5.  99  is  100  —  1.     Hence  to  multiply  a  number  by 
99  we  multiply  it  by  100  and  then  subtract  the  num- 
ber itself  from  the  product.     67  x  99  =  6700  —  67  = 
6633. 

Multiply  34,  76,  29,  and  34  by  99. 

6.  Multiply  45,  63,  72,  and  51  by  999. 

7.  Multiply  32,  24,  48,  and  52  by  98;  also  by  102. 

8.  What  change  is  made  in  the  product  of  28  multi- 
plied by  32  by  adding  10  to  the  multiplier  ?     What 
change  by   adding   10  to  the  multiplicand  ?     What 
change  by  multiplying  the  multiplicand  by  10  ?  the 
multiplier  by  10?    by  multiplying  the  multiplier  and 
multiplicand  both  by  10  ?  by  multiplying  the  multi- 
plicand by  10  and  dividing  the  multiplier  by  10  at 
the  same  time  ? 

9.  What  change  is  made  in  the  quotient  of  80  —•  16 
by  dividing  the  dividend  by  5  ?  by  dividing  the  divisor 
by  4?  by  dividing  both  by  4?     What  change  is  made 
in  the  quotient  by  multiplying  both  the  divisor  and 
dividend  by  4  ? 

10.  What  is  the  square  of  24  ? 

SOLUTION.— 24  =  20  +  4;  (20  +  4)    x    (20  +  4)  gives 


MISCELLANEOUS.  159 

20  x20  +  4x20+4  x  20  +  4x4  =  28x20  +  16=  560  + 
16  =  576.  Hence,  to  square  a  number  of  two  figures 
we  add  the  number  of  units  to  the  number,  multiply 
this  l>y  the  tens,  and  add  to  it  the  square  of  the  units. 
Thus,  to  square  46,  add  6  to  46,  giving  52.  Multiply 
52  l>y  4  tens  or  40,  giving  2080,  and  to  this  add  36. 

11.  What  is  the  square  of  13?   of  14  ?    of  15  ?    16  ? 
IT?  18?  19? 

12.  What  is  the  square  of  20  ?  of  30  ?  of  40  ?  of  90  ? 

13.  Find  the  square  of  31;  of  37;  of  52;  of  56  ;  27. 
In   all   the   preceding    chapters    of    miscellaneous 

questions  the  pupil  should  be  required  to  frame 
several  questions  of  each  sort  presented,  and  the 
teacher  should  add  questions  in  each  recitation. 


160  NUMERATION   AND   NOTATION. 


CHAPTER    XII. 

ELEMENTARY    LESSOXS    IN    WRITTEN 
ARITHMETIC. 

LESSON    I. 
Numeration  and  Notation. 

1.  IN  the  preceding  chapters  the  pupil  has  learned 
orally  the  numeration   and   notation  of  numbers   as 
high   as  thousands.     The  numbers  from  one  to  one 
thousand    form    a  class  of  three  orders  of  numbers. 
The   first  order  is  units;  second,  tens  of  units;   and 
third,  hundreds  of  units.     In  the  same  manner  there 
are  three  orders  of  the  class  thousands:  first,  thou- 
sands ;  second,  tens  of  thousands  •  third,  hundreds  of 
thousands.     As  to  ten  hundred  we  gave  a  new  name, 
thousand  •   so  to  ten  hundred  thousand  it  is  conve- 
nient to  give  a  new  name,  and  thus  form  a  third  class. 
This  we  call  a  'million. 

2.  We  have  also  learned  in  a  former  chapter  how 
to  express  numbers  as  high  as  thousands  by  means  of 
the  nine  Arabic  or  Indian  figures,  together  with  the 
symbol  of  place,  0.     These  nine  figures  are  called  sig- 
nificant figures^  and  0  is  called  a  cipher,  or  naught,  or 
zero.     We  have  learned  that  a  figure,  as  7,  standing 
by  itself  expresses  7  simple  units ;  that  a  0  or  another 
figure  placed  to  the  right  of  it,  thus,  70  or  72,  makes 
it  express  seven  tens,  that  is,  seven  units  of  the  next 
higher  order;  that  two  O's,  or  two  figures,  or  a  cipher 
and  a  significant  figure,  placed  to  the  right  of  it,  as 
700,  708,  718,  or  780,  make  it  express  seven  units  of 


NUMERATION    AND   NOTATION.  161 

the  next  higher  order,  or  seven  hundreds ;  and  that 
three  significant  figures  or  ciphers  placed  to  the  right 
of  it  cause  the  seven  to  express  seven  thousands. 

3.  The  law  here  is,  that  each  additional  figure  to  the 
right  of  a  figure  increases  tenfold  the  value  of  the 
original  figure.  We  can  now  extend  this  law  to  all. 
higher  numbers.  To  make  the  figure  7  express  seven 
tens  of  thousands,  we  annex  four  figures  to  it ;  to  make 
it  express  seven  hundreds  of  thousands,  we  annex  five 
figures ;  to  make  it  express  seven  millions,  we  annex 
six  figures.  If  a  number  is  expressed  in  seven  figures, 
the  first  figure  on  the  right  expresses  units,  the  next 
tens,  the  next  hundreds,  the  next  thousands,  the  next 
tens  of  thousands,  and  so  on  to  millions.  That  is, 
moving  a  figure  one  place  to  the  left,  makes  its  value 
ten  times  as  great  as  before. 

Table. 

•£ 


PI 

c3 

00 

g 

•^ 

•s 

§ 

«w 

2 

o 

o 

.      -3 
1      | 

•£ 

"o 

1 

CO 

CO 

3      s 

r3        d 

ce 

d 

n 

o 

3 

^     M 

H 

EH 

PH 

I, 

3 

3 

5, 

6 

2 

9 

o, 

1 

4,      9 

0 

2, 

6 

to 


7  4 

4  1  (2' 

6  2  (3) 

0  0  (4) 

(1)  iri  read,  One  thousand  three  hundred  and  seventy- 
four. 


162  NUMERATION   AND   NOTATION. 

(2)  is  read,  Thirty-five  thousand  six  hundred  and 
forty-one. 

(3)  is  read,  Two  hundred  and  ninety  thousand  one 
hundred  and  sixty-two. 

(4)  is  read,  Four  million   nine  hundred   and  two 
thousand  six  hundred. 

4.  The  0  is  put  in  the  place  of  any  order  of  units 
which  is  wanting  in  the  number  to  be  expressed. 
Thus  five  thousand  and  six  has  no  tens  nor  hundreds. 
Hence  it  is  written  5006,  because  in  order  that  5  may 
express  five  thousand,  it  must  have  three  figures  after 
it,  and  as  we  have  only  one  significant  figure  to  write, 
we  therefore  use  two  O's. 

NOTE. — When  the  number  of  tens  is  more  than  1,  we  use  the  ter- 
mination -ty.  74  is  seven- ty-four,  instead  of  seven-ten s-and-four. 
Twenty,  thirty,  forty,  fifty,  are  slightly  changed  from  twain-ty,  three-ty, 
four-ty,  five-ty.  In  62,354,  instead  of  saying  six-tens-and-two  thou- 
sand three  hundred  five-tens-and-four,  we  say  sixty-two  thousand 
three  hundred  and  fifty-four. 

5.  EXERCISES. 

Let  the  pupil  read,  or  write  in  words,  the  following 
numbers : 

1.  684,    596,   4000,   24000,   1000,   1001,    1021. 

2.  50,    500,    5000,    50000,    50500,   50505. 

3.  403,   430,   4308,   4764,   40764. 

4.  10000,    10094,   10904,   10094. 

5.  60,    604,   1604,   21604,   421640. 

6.  123,    321,   312,   444,   4444,   44444,   444444. 

7.  2000,   20000,   200200,   2200200,   2220202. 
Express  in  figures  the  following  numbers : 

1.  Six  hundred;  six  hundred  and  forty;  six  hun- 
dred and  forty-six. 

2.  Four  hundred  and  four ;  four  hundred  and  four- 
teen ;  four  hundred  and  forty.  v 

3.  Three  hundred  and  thirty ;  three  thousand  ;  three 
thousand  and  three  hundred. 


ADDITION.  163 

4.  Six  thousand;  six  thousand  four  hundred  and 
forty  ;  six  thousand  four  hundred  and  four. 

5.  Nine  hundred  and  one ;  one  hundred  and  nine  ; 
one  hundred  and  ninety. 

6.  Eight  hundred ;  eight  Hundred  thousand ;  eight 
hundred  thousand  four  nundred  and  fifty ;  four  hun- 
dred and  fifty-five. 

7.  One   thousand ;   one   million ;   six   million ;   six 
million  six  thousand  and  six  hundred ;  six  million  six 
hundred  and  fifty-four  thousand  three  hundred  and 
twelve. 

LESSON    II. 
Addition. 

6.  By  ADDITION  we  unite  two  or  more  numbers  in 
one  number. 

The  result  of  the  addition  is  called  the  SUM  or 
AMOUNT. 

In  adding  large  numbers,  we  add  the  units,  the 
tens,  the  hundreds,  &c.,  separately.  Two  cases  occur. 

7.  I.  When  the  sum  of  each  column  is  not  more 
than  9. 

EXAMPLE  1.— Add  641,  132,  213. 

Operation. 
641 
132 
213 

986 

We  write  the  numbers  so  that  units  are  in  the  same 
column,  tens  in  the  same  column,  hundreds  in  the 
same  column.  Beginning  with  units,  we  add :  3  and 
2  and  1  are  6.;  1  and  3  and  4  are  8 ;  2  and  1  and  6 
are  9.  The  sum  is  986. 


164:  ADDITION. 

Add  the  following  numbers : 

2.  234         2222         3333         2000  5001  4210 

312         3132         2141         1423  2111  1434 

421         2321         2203         1004  422  1244 

22         2314         1221         4352  1241  3101 


3.  What  is  the  sum  of  3241,  1320,  2103,  1025  ? 

4.  Add  together  2345,  1201,  4321,  1102. 

5.  Add  together  1234,  4321,  2222,  2222. 

6.  A  man  bought  a  house  for  2300  dollars,  a  horse 
for  250  dollars,  and  a  wagon  for  120  dollars;  what 
did  he  pay  for  them  all  ? 

8.  II.  When  the  amount  in  any  column  is  more 
than  9. 

EXAMPLE. — If  a  merchant  has  1276  dollars  in  bank 
bills,  479  dollars  in  gold,  784  dollars  in  silver,  and  4 
dollars  in  cents,  how  much  money  has  he  in  all  ? 

Operation. 
1276 
479 

784 
4 


2543 

We  write,  as  before,  units  under  units,  tens  under 
tens,  &c.  4,  4,  9  and  6  units  are  23  units,  or  2  tens 
and  3  units.  We  write  3  units  tinder  units,  and  add 
the  2  tens  with  the  column  of  tens,  thus :  2,  8,  7  and 
7  are  24  tens,  or  2  hundreds  and  4  tens.  Writing  4 
tens  under  tens,  we  add  2  hundreds  with  the  column 
of  hundreds  :  2,  7,  4  and  2  are  15  (hundred).  We  write 
5  under  hundreds,  and  add  1  with  thousands :  1  and  1 
are  2.  The  sum  is  2543.  The  merchant  has  2543 
dollars. 


ADDITION.  165 

EXAMPLES. 

1.      2704         4623             66  704  358 

48502           598           749  8506  4854 

79508         1942         8515  9402  987 

706           749        15652  56708  74749 


2.490  777  333  888  888  123 

244  444  330  22  99  221 

130  999  444  3  11  456 

355  442  440  44  2  654 

240  111  505  555  33  789 

166  222  606  6  590  978 


3.  8649  9876  5436  6  6789 

5842  5432  4563  66  9876 

3876  1234  3654  6836  7890 

9204  5678  5463  6956  9876 

7587  1875  6345  6796  1234 


4.  How  many  are  902010  +  178393  +  643421  + 
366780  +  703020  +  683463  +  604050  +  221894? 

5.  What  is  the  sum  of  forty-four   thousand  three 
hundred  and  twelve ;  six  thousand  eight  hundred  and 
twenty-seven  ;   nine   hundred   and   fifty -.two  ;    thirty 
thousand  and  thirty  ? 

6.  Find  the  sum  of  ninety ;  eighty  thousand  seven 
hundred  and  one ;  two  thousand  seven  hundred  and 
fifty-four;  one  thousand  five  hundred  and  seventy; 
forty-two  thousand  and  four. 

7.  In  a  nursery  of  trees  there  are  1250  apple  trees, 
2050  pear  trees,  3500  peach  trees,  1750  plum  trees, 
850  apricot  trees;  how  many  trees  in  all? 


166  SUBTRACTION. 

LESSON    III. 

Subtraction. 

9.  By  SUBTRACTION  we  find  the  difference  of  two 
numbers,  by  taking  the  less  number  from  the  greater. 

The  greater  number  is  the  MINUEND. 

The  less  number  is  the  SUBTRAHEND. 

The  difference  is  also  called  the  REMAINDER. 

To  subtract,  we  place  the  subtrahend  under  the 
minuend  ;  units  under  units,  tens  under  tens,  &c. 
Two  cases  may  arise. 

10,  I.  Each  figure  in  the  subtrahend  may  be  less 
than  the  figure  above  it  in  the  minuend. 

EXAMPLE  1.  —  From  576  subtract  324. 

Operation. 
576 


252 

Subtracting  each  figure  in  the  subtrahend  from  the 
figure  above  it,  beginning  with  units,  we  have  4  from 
6,  2  ;  2  from  7,  5  ;  3  from  5,  2.  The  remainder  is  252. 

PROOF.  —  Add  the  units,  tens,  &c.  of  the  subtrahend 
and  remainder  —  2  and  4  are  6;  5  and  2  are  7;  2  and 
3  are  5  —  and  we  have  the  minuend,  576. 

EXAMPLES. 

2.  From  540  take  200.  From  860  take  300. 

3.  800  5000    60000    9000    790   80000 
200  2000    20000    3000    400    3000 


4.  8760      9630         6320        5670      4810        5945 
3000       7000         2000         3000       1000         3000 


SUBTRACTION.  167 

5.  937        877      697      867      633       857       6848 
122         352       245       423       121       307        3007 


6.  85674         4789          62476          34897 
1352          3254  231          12244 


11.  II.  One  or  more  figures  in  the  subtrahend  may 
be  greater  than  the  figures  above  them  in  the  minu- 
end. 

EXAMPLE. — From  468  take  349. 

Operation. 

10 

468 
349 

119 

Write  the  less  number  under  the  greater.  As  we 
cannot  take  9  units  from  8  units,  we  mentally  add  10 
to  8,  and  then  say,  9  from  18  leaves  9,  which 'we  write 
down.  Since  we  added  10  to  the  minuend,  we  add  1 
ten, to  the  4  tens  of  the  subtrahend,  and  say  5  from  6 
leaves  1,  which  we  write  down.  Finally,  3  hundred 
from  four  hundred  leaves  1  hundred.  119  is  the  dif- 
ference between  the  numbers,  because  we  have  taken 
all  the  parts  of  the  less  from  the  greater,  and  we  have 
balanced  the  ten  added  to  the  greater  by  adding  ten 
to  the  less. 

EXAMPLE. — From  9434  take  3537. 

Operation. 

101010 

9434 
3537 

5897 


168 


SUBTRACTION. 


We  say,  7  from  14,  7 ;  4  from  13,  9 ;  G  from  14,  8  ; 
4  from  9,  5.  The  remainder  is  5897. 

If  there  are  fewer  figures  in  the  lower  line  than  in 
the  upper,  proceed  as  if  O's  were  written  on  the  left  of 
the  lower  number  until  the  number  of  figures  is  the 
same  in  both. 


732 
13 


807 
718 


8403 
2390 


EXAMPLES. 

10 

1. 

670 

860   490 

540 

650   880 

25 

43    38 

26 

39    32 

10 

10 

2. 

7860 

1780 

2690 

806 

709 

243 

437 

272 

243 

333 

10 

3. 

6072 

7043 

4068 

5037 

2342 

2520 

3660 

1717 

10      10  101010- 

4.  4372     782       965       585       7184      4360      7343 
2657     355      428       247       6526       3608       6740 


1715 

5.  2814    375    5006    4176    7583   8064 
1033    185    2818    3648-    3065   3975 


6.  6004   60010    10000    8200102    815240 
3716     528     3196    4866769    379468 


7.  7802008    10000000    951315     9555555 
3357564         1     947688    2888889 


MULTIPLICATION.  169 

8.  How  much  smaller  is  38567  than  60600  ? 

9.  How  much  is  4009000  less  5678  ? 

10.  How  much  greater  is  700000  than  599328  ? 

11.  Take  from  43518  the  number  7253,  and  from 
the  remainder  7253,  and  so  on,  and  give  the  remainder 
after  the  sixth  subtraction. 

12.  How  much  is  3820  +  16488  +  347  +  5688  + 
2433  -  31210? 

13.  A  man  owns  a  farm  which  cost  him  5500  dol- 
lars ;  he  has  paid  2875  dollars  ;  how  much  more  has 
he  to  pay. 

14.  A  merchant  sold  for  2825  dollars  goods  which 
cost  2138  dollars ;  how  much  did  he  gain  ? 

15.  A  man  bought  a  house  and  lot  for  2750  dollars, 
and  sold  it  for  3125  dollars ;  how  much  did  he  gain  ? 

16.  How  old  would  a  man  be  in  1868  who  was 
born  in  1839? 

LESSON    IV. 

Jfuttiplicat  ion. 

12.  If  a  man  can  earn  8  dollars  in  one  week,  in  4 
weeks  he  can  earn  4  times  8  dollars,  or  32  dollars. 

This  is  multiplication:  8,  the  number  to  be  re- 
peated, is  called  the  MULTIPLICAND  :  4,  which  shows 
how  many  times  the  multiplicand  is  to  be  taken,  is 
called  the  MULTIPLIER,  and  32,  the  result  of  the  multi- 
plication, is  the  PRODUCT. 

In  multiplying  large  numbers,  there  are  two  cases : 

13.  I.  When  the  multiplier  is  a  single  figure. 
EXAMPLE  1. — Multiply  467  by  5. 

Operation. 

467 
5 

2335 


170  MULTIPLICATION. 

.  We  write  the  multiplier  under  the  multiplicand, 
units  under  units,  and  say,  5  times  7  units  are  35 
units,  or  three  tens  and  5  units.  We  write  5  units 
below,  and  reserve  3  tens.  5  times  6  tens  are  30  tens, 
which  with  3  tens  make  33  tens,  or  3  hundreds  and 
3  tens.  Writing  3  tens  in  tens'  place,  we  reserve  3 
hundred.  5  times  4  hundred  are  20  hundred,  which 
with  3  hundred  mate  23  hundred.  This  we  write 
down,  and  have  for  the  product,  2335. 

2.  356          825         6745         4567  Multiplicand. 

4536  Multiplier. 

1424  27402  Product. 

3.  8474  9876  5432  12345 

2789 


4.  Multiply  4825  by  2;  by  3 ;  by  4;  by  5 ;  by  6. 

5.  Multiply  3775  by  5 ;  by  6 ;  by  7 ;  by  8 ;  by  9. 

6.  Multiply  5709  by  4;  'by  5;  by  6;  by  7;  by  8; 
by  9. 

7.  If  one  horse  costs  275  dollars,  what  is  the  cost 
of  8  horses  ? 

8.  If  one  house  is  worth  3275  dollars,  what  is  the 
value  of  7  similar  houses  ? 

9.  If  the  multiplicand  is  3917  and  the  multiplier 
is  6,  what  is  the  product? 

10.  Multiply  the  sum  of  1847  and  2043  by  8. 

11.  Multiply  the  difference  between  3194  and  1536 
by  9. 

14.  II.  When  the  multiplier  contains  more  than 
one  figure. 

EXAMPLE  1.— Multiply  843  by  26. 


MULTIPLICATION.  171 

Operation,, 

843 

26 


5058 
1686 

21918 

Write  the  multiplier  under  the  multiplicand,  units 
under  units,  tens  under  tens,  &c. 

Multiply  by  6,  the  units  figure,  as  in  Case  I.  In 
multiplying  by  2  tens,  place  the  first  figure  of  this 
product  under  2,  the  multiplier.  For,  2  tens  times  3 
are  6  tens;  therefore,  the  6  should  be  in  the  second, 
or  tens',  place. 

Add  the  two  partial  products,  and  find  the  entire 
product,  21918. 

EXAMPLE  2.  Multiply  4357  by  345. 

Operation. 

4357 
345 


21785 
17428 
13071 

1503165 

7  _  35  ;  write  5  and  "  carry"  3. 

=  25,  +  3  —  28  ;  write  8. 
=  15,  +  2  =  17 ;  write  7. 
=  20,  +  1  =  21 ;  write  21. 

7  =  28  ;  write  8  under  4. 
5  =  20,  +  2  —  22  ;  write  2. 

3  _  12,  +  2  =  14  ;  write  4. 

4  =  16,  +  1  —  17 ;  write  17. 


1 72  MULTIPLICATION. 

3  x  7  =  21 ;  write  1  under  3. 
3  x  5  =  15,  +  2  =  17 ;  write  7. 
3x3  =  9,  +  1  =  10;  write  0. 
3  x  4  —  12,  +  1  —  13  ;  write  13. 

The  sum  of  the  partial  products  is  the  entire  pro- 
duct, 1,503,165. 

3.  3458  8431  1357          2465          5438 

24  35  54  89  146 


4.  5417          5104          3547      .    1596          2705 
123  456  258  357  403 


5.  Multiply  1047  by  284;  by  207;  by  615. 

6.  Multiply  3579  by  123 ;  by  612  ;  by  246. 

7.  Multiply  4680  by  340 ;  by  360  ;  by  801. 

8.  Multiply  6143  by  200;  by  4200. 

6143  6143 

200  4200 


1228600  1228600 

24572 


25800600 

9.  Multiply  3434  by  300  ;  by  5300. 

10.  Multiply  1234  by  600  ;  by  1054. 

11.  1417  x  34;  6417  x  27;  6946  x  123. 

12.  340  x  20  x  16 ;  853  x  300  x  4 ;  803  x  101. 

NOTE. — The  teacher  will  readily  supply  additional  examples  for 
practice  under  each  rule.  They  can  be  written  on  the  blackboard, 
and  copied  by  the  pupils. 


DIVISION.  173 

LESSON    V. 
Division. 

15.  By  DIVISION  we  find  how  many  times  one  num- 
ber is  contained  in  another  larger  number. 

The  DIVIDEND  is  the  number  to  be  divided. 

The  DIVISOR  is  the  number  by  which  we  divide. 

The  number  of  times  the  divisor  is  contained  in  the 
dividend  is  the  QUOTIENT. 

The  REMAINDER  is  what  is  sometimes  left  after 
dividing. 

16.  There  are  two  methods  of  performing  division. 
Ex.  1.  How  many  times  is  4  contained  in  948  ? 

1st  Operation. 

4)948(237 
8 

14 
12 

28 

28 

0 

We  begin  at  the  left  of  the  dividend  to  divide.  4 
is  contained  in  9,  2  times.  Place  2  at  the  right  of  the 
dividend,  as  the  first  figure  of  the  quotient.  2  times 
4  are  8,  which  we  set  under  9,  and  subtract.  To  the 
remainder,  1,  we  bring  down  4,  the  next  figure  of  the 
dividend,  making  14.  4  in  14,  3  times.  Place  3  in 
the  quotient  and  multiply  the  divisor,  and  proceed  as 
before,  till  all  the  figures  are  brought  down  and 
divided.  The  entire  quotient  is  237. 


174  DIVISION. 

2d  Operation. 

4)948 

237 

In  this  method,  we  do  in  the  mind  a  part  of  what 
we  wrote  out  in  the  1st  operation,  and  write  the  quo- 
tient under  the  dividend. 

4  is  contained  in  9,  2  times  and  1  over.  We  place 
2  under  9  as  the  first  figure  of  the  quotient,  and 
imagine  the  remainder  1  placed  before  4,  the  next 
figure  of  the  dividend,  making  14.  4  in  14,  3  times 
and  2  over.  We  place  3  under  4,  and  imagine  2  set 
before  8,  making  28.  4  in  28,  7  times.  The  entire 
quotient  is  237,  as  before. 

1 7.  This  second  method  is  sometimes  called  SHORT 
DIVISION,  and  is  used  when  the  divisor  is  not  more 
than  12. 

The  first  method  is  called  LONG  DIVISION,  and  is 
used  when  the  divisor  is  more  than  12. 

2.  9)1251  9)1251(139  7)28345(4049 

9  28 

139  — 

35  34 

27  28 

81  65 

3.  7)28345  81  63 


4049  2  Eem.       0  2  Eem. 

Work  the  following  examples  by  Short  Division : 
4.  4)57432         6)31740         8)17344^       9)5562 

618 


DIVISION.  175 

5.  5)74035      4)67434      6)10746       8)84755 

10594—3  Kern. 

Prove  all  the  examples  in  Division,  by  multiplying 
the  quotient  and  divisor  together  and  adding  the  re- 
mainder, if  there  is  any,  to  the  product.  The  result 
must  equal  the  dividend. 

"Work  the  following  by  both  methods : 

6.  6)74789  7)78743          8)97145         9)23474 


EXAMPLE  7. — Divide  9274  by  35.     Here,  of  course, 
we  use  Long  Division. 

Operation.  Proof. 

35)9274(264  264 

70  35 


227  1320 
210  792 
34 

174 

140  9274 

34  Kem. 

How  many  times  is  35  contained  in  92?  3  in  9,  3 
times;  but,  multiplying  35  by  3,  we  have  105,  which 
is  more  than  92;  we  therefore  take  a  quotient  figure 
one  less,  2,  and  find  2  times  35  =  70,  which  is  less  than 
92.  So,  to  find  the  second  quotient  figure :  3  in  22,  7 
times.  Finding  7  too  large,  we  try  6. 

8.  Divide  3754  by  24.     Quo.  156.     Kem.  10. 

9.  Divide  17473  by  56.     Quo.  312*     Rem.  1. 


176  DIVISION. 

10.  Divide  3746  by  36.     How  can  you  know  whe- 
ther your  quotient  is  correct  ? 

11.  Divide  2756  by  48.  Operation. 
The  divisor,  48,  is  nearly  50.     To    48)2756(57 

find  the  quotient  figures  we  say,  5  in  240 

27,  5  times ;  multiplying  the  divisor 

by  5,  we  have  240,  less  than  275  ;  356 

subtracting,  we  find  35  for  remain-  336 

der,  a  number  less  than  the  divisor. 

So,  5  in  35,  7  times,  &c.  20  Rem. 

12.  Divide  9437  by  27;  by  31 ;  by  35  ;  by  14. 

13.  Divide  10743  by  45  ;  by  450 ;  by  412. 

14.  Divide  12345  by  103  ;  by  301 ;  by  345. 

15.  Since  there  are  365  days  in  one  year,  how  many 
years  are  there  in  2000  days  ? 

16.  If  I  pay  9744  dollars  for  112  acres  of  land,  how 
much  is  it  an  acre  ? 

17.  How  many  cows  can  I  buy  for  581  dollars,  if 
each  cow  is  worth  83  dollars  ? 

18.  If  the  dividend  is  7839,  and  the  divisor  is  214, 
what  are  the  quotient  and  remainder  \ 

19.  Add  together  359,  895,  and   1054;    subtract 
1343  from  the  sum  ;  multiply  the  remainder  by  8, 
and  divide  the  product  by  4.     What  is  the  quotient  ? 

NOTE. — The  same  example    expressed  by  means  of  signs  is  359  -f 
895  +  1054  —  1343  x   8  -j-  4  =  ? 

20.  874  +  917-1056    x  6  ~  5  —  ?      Quotient 
882. 

21.  3846  +  1904  +  3841  —  3489  x  100  ~  75  =  ? 

22.  804  x  200  -f-  25  +  3568  -  10000  =  2 


- 


t 


/ 

£ 


YB   I744P 


7 


8- 


4 


M30G004 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


7     / 


V  E  X  A.  B  L  K  •  s 
MATHEMATICAL 


1.  First    Lessons     in     Numbers.— An    illustrat 

Mental   Arithmetic,  for  the  use  of  Farni; 
Adapt  i .-d  to  the  OBJECT  METHOD.     30  cents. 

2.  Mental  Arithmetic.— After  the  inductive 

ms    in     Written  Arithmetic.      I 
Academies.     45  cents. 

3.  Elements  of  Written  Arithmetic.— In  which  the  fun  dan 

tioTis  on  Abstract  and  Denominate  Numbers,  and  i 

cation*,  are  thoroughly  explained  and  illustrated   by*immer 

amples,  adapted  to  the  business  of  practical  life.      §1, 00. 

4.  High  School  Arithmetic.  FOR  THE  USE  OF  SCHOOLS  AND  COLLEGES.- 


tg  Ol 
lllU! 


on  the  fundamental  principles 
strated   by  a  greater  variety  and  num- 


this  v  ork  the  reaso 
developed,  an 

locations  an  -.     In  addition  to  the  subjects  usually 

rithnietic,  it  will  contain  several  cinq 
ai  bers,  Progressions,  .Logarith ;  i 


5.  Elementary  Algebra.- 

For  Schools  an 

6.  Higher  Algebra 

hapters  on  the  Theory  of  Determinants,  and  othf 
sons  from  the  modern  higher  Algebra,  for  the  use  of  Univ< 
Students. 

7.  Le«^sndre's  Geometry.—  To  the  complete  translation  of  the  i 

ot  *he  standard  works  of  Legendre  will  be  appended  ma- 
in Geometrical   Analysis,  with  hints   for  the  soluti 
Also,  notes  on  the  application  of  the  method  of  Limits  to  ! 
stnitiuu  of  theorems  in  Elementary  Geometry. 

8.  Analytical  Geometry.—  PART  FIRST.—  Plane  Co-ordin 

Point.  Straight  Line,  Conic  Sections,  some  of  the  liiidH-r  PI:. 
'•raic  and  Transcendental,     Also,  the  Meth 

.—The  Point  and  Straight  Line  in  S; 
Surfaces  of  the  Second   Order,    Cones,    Cyliin 
n,    Conoids,  Helicoid- 

Both   parts  will  <• 


RICHARDSON   Sc    COMPANY, 

Publishers    New  York. 

: 


